6.5: Logarithmic Properties
- Use the product rule for logarithms.
- Use the quotient rule for logarithms.
- Use the power rule for logarithms.
- Expand logarithmic expressions.
- Condense logarithmic expressions.
- Use the change-of-base formula for logarithms.
In chemistry, the pH scale is used as a measure of the acidity or alkalinity of a substance. Substances with a pH less than \(7\) are considered acidic, and substances with a pH greater than \(7\) are said to be alkaline. Our bodies, for instance, must maintain a pH close to \(7.35\) in order for enzymes to work properly. To get a feel for what is acidic and what is alkaline, consider the following pH levels of some common substances:
- Battery acid: \(0.8\)
- Stomach acid: \(2.7\)
- Orange juice: \(3.3\)
- Pure water: \(7\) at \(25^\circ C\)
- Human blood: \(7.35\)
- Fresh coconut: \(7.8\)
- Sodium hydroxide (lye): \(14\)
To determine whether a solution is acidic or alkaline, we find its pH, which is a measure of the number of active positive hydrogen ions in the solution. The pH is defined by the following formula, where \([\ce{H^{+}}]\) is the concentration of hydrogen ions in the solution
\[\begin{align} {pH}&=−{\log}([\ce{H^{+}}]) \label{eq1} \\[4pt] &={\log}\left(\dfrac{1}{[\ce{H^{+}}]}\right) \label{eq2} \end{align}\]
The equivalence of Equations \ref{eq1} and \ref{eq2} is one of the logarithm properties we will examine in this section.
Using the Product Rule for Logarithms
Recall that the logarithmic and exponential functions “undo” each other. This means that logarithms have similar properties to exponents. Some important properties of logarithms are given here. First, the following properties are easy to prove.
\[ \begin{align*} \log_b1 &=0 \\[4pt] \log_bb &=1 \end{align*}\]
For example, \({\log}_51=0\) since \(5^0=1\). And \({\log}_55=1\) since \(5^1=5\).
Next, we have the inverse property.
\[ \begin{align*} \log_b(b^x) &=x \\[4pt] b^{\log_b x} &=x, \,x>0 \end{align*}\]
For example, to evaluate \({\log(100)}\), we can rewrite the logarithm as \({\log}_{10}({10}^2)\), and then apply the inverse property \({\log}_b(b^x)=x\) to get \({\log}_{10}({10}^2)=2\).
To evaluate \(e^{\ln(7)}\), we can rewrite the logarithm as \(e^{{\log}_e7}\), and then apply the inverse property \(b^{{\log}_bx}=x\) to get \(e^{{\log}_e 7}=7\).
Finally, we have the one-to-one property.
\[ \log_bM = \log_bN \text{ if and only if } M=N\]
We can use the one-to-one property to solve the equation \({\log}_3(3x)={\log}_3(2x+5)\) for \(x\). Since the bases are the same, we can apply the one-to-one property by setting the arguments equal and solving for \(x\):
\(3x=2x+5\) Set the arguments equal.
\(x=5\) Subtract \(2x\).
But what about the equation \({\log}_3(3x)+{\log}_3(2x+5)=2\)? The one-to-one property does not help us in this instance. Before we can solve an equation like this, we need a method for combining terms on the left side of the equation.
Recall that we use the product rule of exponents to combine the product of exponents by adding: \(x^ax^b=x^{a+b}\). We have a similar property for logarithms, called the product rule for logarithms , which says that the logarithm of a product is equal to a sum of logarithms. Because logs are exponents, and we multiply like bases, we can add the exponents. We will use the inverse property to derive the product rule below.
Given any real number \(x\) and positive real numbers \(M\), \(N\), and \(b\), where \(b≠1\), we will show
\({\log}_b(MN)={\log}_b(M)+{\log}_b(N)\).
Let \(m={\log}_bM\) and \(n={\log}_bN\). In exponential form, these equations are \(b^m=M\) and \(b^n=N\). It follows that
\[\begin{align*} {\log}_b(MN)&= {\log}_b(b^mb^n) \qquad \text{Substitute for M and N}\\[4pt] &= {\log}_b(b^{m+n}) \qquad \text{Apply the product rule for exponents}\\[4pt] &= m+n \qquad \text{Apply the inverse property of logs}\\[4pt] &= {\log}_b(M)+{\log}_b(N) \qquad \text{Substitute for m and n} \end{align*}\]
Note that repeated applications of the product rule for logarithms allow us to simplify the logarithm of the product of any number of factors. For example, consider \({\log}_b(wxyz)\). Using the product rule for logarithms, we can rewrite this logarithm of a product as the sum of logarithms of its factors:
\({\log}_b(wxyz)={\log}_bw+{\log}_bx+{\log}_by+{\log}_bz\)
The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.
\[\begin{align} {\log}_b(MN)={\log}_b(M)+{\log}_b(N)\text{ for } b> 0 \end{align}\]
- Factor the argument completely, expressing each whole number factor as a product of primes.
- Write the equivalent expression by summing the logarithms of each factor.
Expand \({\log}_3(30x(3x+4))\).
Solution
We begin by factoring the argument completely, expressing \(30\) as a product of primes.
\({\log}_3(30x(3x+4))={\log}_3(2⋅3⋅5⋅x⋅(3x+4))\)
Next we write the equivalent equation by summing the logarithms of each factor.
\({\log}_3(30x(3x+4))={\log}_3(2)+{\log}_3(3)+{\log}_3(5)+{\log}_3(x)+{\log}_3(3x+4)\)
Expand \({\log}_b(8k)\).
- Answer
-
\({\log}_b2+{\log}_b2+{\log}_b2+{\log}_bk=3{\log}_b2+{\log}_bk\)
Using the Quotient Rule for Logarithms
For quotients, we have a similar rule for logarithms. Recall that we use the quotient rule of exponents to combine the quotient of exponents by subtracting: \(x^{\frac{a}{b}}=x^{a−b}\). The quotient rule for logarithms says that the logarithm of a quotient is equal to a difference of logarithms.
The quotient rule for logarithms can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms.
\[{\log}_b\left(\dfrac{M}{N}\right)={\log}_bM−{\log}_bN\]
Just as with the product rule, we can use the inverse property to derive the quotient rule.
Given any real number \(x\) and positive real numbers \(M\), \(N\), and b, b, where \(b≠1\), we will show
\({\log}_b\left(\dfrac{M}{N}\right)={\log}_b(M)−{\log}_b(N)\).
Let \(m={\log}_bM\) and \(n={\log}_bN\). In exponential form, these equations are \(b^m=M\) and \(b^n=N\). It follows that
\[\begin{align*} {\log}_b\left (\dfrac{M}{N} \right )&= {\log}_b\left(\dfrac{b^m}{b^n}\right) \qquad \text{Substitute for M and N}\\[4pt] &= {\log}_b(b^{m-n}) \qquad \text{Apply the quotient rule for exponents}\\[4pt] &= m-n \qquad \text{Apply the inverse property of logs}\\[4pt] &= {\log}_b(M)-{\log}_b(N) \qquad \text{Substitute for m and n} \end{align*}\]
For example, to expand \({\log}\left (\dfrac{2x^2+6x}{3x+9} \right )\), we must first express the quotient in lowest terms. Factoring and canceling we get,
\[\begin{align*} {\log}\left (\dfrac{2x^2+6x}{3x+9} \right )&= {\log}\left (\dfrac{2x(x+3)}{3(x+3)} \right ) \qquad \text{Factor the numerator and denominator}\\[4pt] &= {\log}\left (\dfrac{2x}{3} \right ) \qquad \text{Cancel the common factors} \end{align*}\]
Next we apply the quotient rule by subtracting the logarithm of the denominator from the logarithm of the numerator. Then we apply the product rule.
\[ \begin{align*} {\log}\left(\dfrac{2x}{3}\right) &={\log}(2x)−{\log}(3) \\[4pt] &={\log}(2)+{\log}(x)−{\log}(3) \end{align*}\]
- Express the argument in lowest terms by factoring the numerator and denominator and canceling common terms.
- Write the equivalent expression by subtracting the logarithm of the denominator from the logarithm of the numerator.
- Check to see that each term is fully expanded. If not, apply the product rule for logarithms to expand completely.
Expand \({log}_2\left(\dfrac{15x(x−1)}{(3x+4)(2−x)}\right)\).
Solution
First we note that the quotient is factored and in lowest terms, so we apply the quotient rule.
\({\log}_2(\dfrac{15x(x−1)}{(3x+4)(2−x)})={\log}_2(15x(x−1))−{\log}_2((3x+4)(2−x))\)
Notice that the resulting terms are logarithms of products. To expand completely, we apply the product rule, noting that the prime factors of the factor 15 are 3 and 5.
\[\begin{align*} {\log}_2(15x(x-1))-{\log}_2((3x+4)(2-x))&= [{\log}_2(3)+{\log}_2(5)+{\log}_2(x)+{\log}_2(x-1)]-[{\log}_2(3x+4)+{\log}_2(2-x)]\\[4pt] &= {\log}_2(3)+{\log}_2(5)+{\log}_2(x)+{\log}_2(x-1)-{\log}_2(3x+4)-{\log}_2(2-x) \end{align*}\]
Analysis
There are exceptions to consider in this and later examples. First, because denominators must never be zero, this expression is not defined for \(x=−\dfrac{4}{3}\) and \(x=2\). Also, since the argument of a logarithm must be positive, we note as we observe the expanded logarithm, that \(x>0\), \(x>1\), \(x>−\dfrac{4}{3}\), and \(x<2\). Combining these conditions is beyond the scope of this section, and we will not consider them here or in subsequent exercises.
Expand \({\log}_3\left(\dfrac{7x^2+21x}{7x(x−1)(x−2)}\right)\).
- Answer
-
\({\log}_3(x+3)−{\log}_3(x−1)−{\log}_3(x−2)\)
Using the Power Rule for Logarithms
We’ve explored the product rule and the quotient rule, but how can we take the logarithm of a power, such as \(x^2\)? One method is as follows:
\[\begin{align*} {\log}_b(x^2)&= {\log}_b(x\cdot x)\\[4pt] &= {\log}_bx+{\log}_bx\\[4pt] &= 2{\log}_bx \end{align*}\]
Notice that we used the product rule for logarithms to find a solution for the example above. By doing so, we have derived the power rule for logarithms , which says that the log of a power is equal to the exponent times the log of the base. Keep in mind that, although the input to a logarithm may not be written as a power, we may be able to change it to a power. For example,
\(100={10}^2\) \(\sqrt{3}=3^{\dfrac{1}{2}}\) \(\dfrac{1}{e}=e^{−1}\)
The power rule for logarithms can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.
\[{\log}_b(M^n)=n{\log}_bM\]
- Express the argument as a power, if needed.
- Write the equivalent expression by multiplying the exponent times the logarithm of the base.
Expand \({\log}_2x^5\).
Solution
The argument is already written as a power, so we identify the exponent, 5, and the base, \(x\), and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.
\({\log}_2(x^5)=5{\log}_2x\)
Expand \(\ln x^2\).
- Answer
-
\(2\ln x\)
Expand \({\log}_3(25)\) using the power rule for logs.
Solution
Expressing the argument as a power, we get \({\log}_3(25)={\log}_3(5^2)\).
Next we identify the exponent, \(2\), and the base, \(5\), and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.
\({\log}_3(52)=2{\log}_3(5)\)
Expand \(\ln\left (\dfrac{1}{x^2} \right )\).
- Answer
-
\(−2\ln(x)\)
Rewrite \(4\ln(x)\) using the power rule for logs to a single logarithm with a leading coefficient of \(1\).
Solution
Because the logarithm of a power is the product of the exponent times the logarithm of the base, it follows that the product of a number and a logarithm can be written as a power. For the expression \(4\ln(x)\), we identify the factor, \(4\), as the exponent and the argument, \(x\), as the base, and rewrite the product as a logarithm of a power: \(4\ln(x)=\ln(x^4)\).
Rewrite \(2{\log}_34\) using the power rule for logs to a single logarithm with a leading coefficient of \(1\).
- Answer
-
\({\log}_316\)
Expanding Logarithmic Expressions
Taken together, the product rule, quotient rule, and power rule are often called “laws of logs.” Sometimes we apply more than one rule in order to simplify an expression. For example:
\[\begin{align*} {\log}_b \left (\dfrac{6x}{y} \right )&= {\log}_b(6x)-{\log}_by\\[4pt] &= {\log}_b6+{\log}_bx-{\log}_by \end{align*}\]
We can use the power rule to expand logarithmic expressions involving negative and fractional exponents. Here is an alternate proof of the quotient rule for logarithms using the fact that a reciprocal is a negative power:
\[\begin{align*} {\log}_b\left (\dfrac{A}{C} \right )&= {\log}_b(AC^{-1})\\[4pt] &= {\log}_b(A)+{\log}_b(C^{-1})\\[4pt] &= {\log}_bA+(-1){\log}_bC\\[4pt] &= {\log}_bA−{\log}_bC \end{align*}\]
We can also apply the product rule to express a sum or difference of logarithms as the logarithm of a product.
With practice, we can look at a logarithmic expression and expand it mentally, writing the final answer. Remember, however, that we can only do this with products, quotients, powers, and roots—never with addition or subtraction inside the argument of the logarithm.
Rewrite \(ln \left (\dfrac{x^4y}{7} \right )\) as a sum or difference of logs.
Solution
First, because we have a quotient of two expressions, we can use the quotient rule:
\(\ln \left (\dfrac{x^4y}{7} \right )=\ln(x^4y)−\ln(7)\)
Then seeing the product in the first term, we use the product rule:
\(\ln(x^4y)−\ln(7)=\ln(x^4)+\ln(y)−\ln(7)\)
Finally, we use the power rule on the first term:
\(\ln(x^4)+\ln(y)−\ln(7)=4\ln(x)+\ln(y)−\ln(7)\)
Expand \(\log \left (\dfrac{x^2y^3}{z^4} \right )\).
- Answer
-
\(2\log x+3\log y−4\log z\)
Expand \(\log(x)\).
Solution
\[\begin{align*} \log(\sqrt{x})&= \log x^{\left (\tfrac{1}{2} \right )}\\[4pt] &= \dfrac{1}{2}\log x \end{align*}\]
Expand \(\ln(\sqrt[3]{x^2})\).
- Answer
-
\(\dfrac{2}{3}\ln x\)
No. There is no way to expand the logarithm of a sum or difference inside the argument of the logarithm.
Expand \({\log}_6 \left (\dfrac{64x^3(4x+1)}{(2x−1)} \right )\).
Solution
We can expand by applying the Product and Quotient Rules.
\[\begin{align*} {\log}_6\left (\dfrac{64x^3(4x+1)}{(2x-1)} \right )&= {\log}_664+{\log}_6x^3+{\log}_6(4x+1)-{\log}_6(2x-1)\qquad \text{Apply the Quotient Rule}\\[4pt] &= {\log}_626+{\log}_6x^3+{\log}_6(4x+1)-{\log}_6(2x-1) \qquad \text{Simplify by writing 64 as } 2^6\\[4pt] &= 6{\log}_62+3{\log}_6x+{\log}_6(4x+1)-{\log}_6(2x-1) \qquad \text{Apply the Power Rule} \end{align*}\]
Expand \(\ln \left (\dfrac{\sqrt{(x−1){(2x+1)}^2}}{(x^2−9)}\right )\).
- Answer
-
\(\dfrac{1}{2}\ln(x−1)+\ln(2x+1)−\ln(x+3)−\ln(x−3)\)
Condensing Logarithmic Expressions
We can use the rules of logarithms we just learned to condense sums, differences, and products with the same base as a single logarithm. It is important to remember that the logarithms must have the same base to be combined. We will learn later how to change the base of any logarithm before condensing.
- Apply the power property first. Identify terms that are products of factors and a logarithm, and rewrite each as the logarithm of a power.
- Next apply the product property. Rewrite sums of logarithms as the logarithm of a product.
- Apply the quotient property last. Rewrite differences of logarithms as the logarithm of a quotient.
Write \({\log}_3(5)+{\log}_3(8)−{\log}_3(2)\) as a single logarithm.
Solution
Using the product and quotient rules
\({\log}_3(5)+{\log}_3(8)={\log}_3(5⋅8)={\log}_3(40)\)
This reduces our original expression to
\({\log}_3(40)−{\log}_3(2)\)
Then, using the quotient rule
\({\log}_3(40)−{\log}_3(2)={\log}_3 \left (\dfrac{40}{2} \right )={\log}_3(20)\)
Condense \({\log}_3−{\log}_4+{\log}_5−{\log}_6\).
- Answer
-
\(\log \left (\dfrac{3⋅5}{4⋅6} \right)\); can also be written \(\log \left (\dfrac{5}{8} \right )\) by reducing the fraction to lowest terms.
Condense \({\log}_2(x^2)+\dfrac{1}{2}{\log}_2(x−1)−3{\log}_2({(x+3)}^2)\).
Solution
We apply the power rule first:
\({\log}_2(x^2)+\dfrac{1}{2}{\log}_2(x−1)−3{\log}_2({(x+3)}^2)={\log}_2(x^2)+{\log}_2(\sqrt{x−1})−{\log}_2({(x+3)}^6)\)
Next we apply the product rule to the sum:
\({\log}_2(x^2)+{\log}_2(\sqrt{x−1})−{\log}_2({(x+3)}^6)={\log}_2(x^2\sqrt{x−1})−{\log}_2({(x+3)}^6)\)
Finally, we apply the quotient rule to the difference:
\({\log}_2(x^2\sqrt{x−1})−{\log}_2({(x+3)}^6)={\log}_2\dfrac{x^2\sqrt{x−1}}{{(x+3)}^6}\)
Rewrite \(2\log x−4\log(x+5)+\dfrac{1}{x}\log(3x+5)\) as a single logarithm.
Solution
We apply the power rule first:
\(2\log x−4\log(x+5)+\dfrac{1}{x}\log(3x+5)=\log(x^2)−\log{(x+5)}^4+\log({(3x+5)}^{x^{−1}})\)
Next we rearrange and apply the product rule to the sum:
\[\begin{align*} \log(x^2)-\log{(x+5)}^4+\log({(3x+5)}^{x^{-1}})&= \log(x^2)+\log({(3x+5)}^{x^{-1}}-\log{(x+5)}^4\\[4pt] &= \log(x^2{(3x+5)}^{x^{-1}})-\log{(x+5)}^4\\[4pt] &= \log\dfrac{x^2{(3x+5)}^{x^{-1}}}{{(x+5)}^4} \qquad \text{Apply the quotient rule to the difference} \end{align*}\]
Rewrite \(\log(5)+0.5\log(x)−\log(7x−1)+3\log(x−1)\) as a single logarithm.
- Answer
-
\(\log \dfrac{5{(x−1)}^3\sqrt{x}}{(7x−1)}\)
Condense \(4(3\log(x)+\log(x+5)−\log(2x+3))\).
- Answer
-
\(\log\dfrac{x^{12}{(x+5)}^4}{{(2x+3)}^4}\); this answer could also be written \(\log{ \left (\dfrac{x^3(x+5)}{(2x+3)} \right )}^4\)
Recall that, in chemistry, \({pH}=−\log[H+]\). If the concentration of hydrogen ions in a liquid is doubled, what is the effect on pH?
Solution
Suppose \(C\) is the original concentration of hydrogen ions, and \(P\) is the original pH of the liquid. Then \(P=–\log(C)\). If the concentration is doubled, the new concentration is \(2C\). Then the pH of the new liquid is
\(pH=−\log(2C)\)
Using the product rule of logs
\(pH=−\log(2C)=−(\log(2)+\log(C))=−\log(2)−\log(C)\)
Since \(P=–\log(C)\),the new pH is
\(pH=P−\log(2)≈P−0.301\)
When the concentration of hydrogen ions is doubled, the pH decreases by about \(0.301\).
How does the pH change when the concentration of positive hydrogen ions is decreased by half?
- Answer
-
The pH increases by about \(0.301\).
Using the Change-of-Base Formula for Logarithms
Most calculators can evaluate only common and natural logs. In order to evaluate logarithms with a base other than \(10\) ore, e,we use the change-of-base formula to rewrite the logarithm as the quotient of logarithms of any other base; when using a calculator, we would change them to common or natural logs.
To derive the change-of-base formula, we use the one-to-one property and power rule for logarithms .
Given any positive real numbers \(M\), \(b\), and \(n\), where \(n≠1\) and \(b≠1\),we show
\({\log}_bM=\dfrac{{\log}_nM}{{\log}_nb}\)
Let \(y={\log}_bM\). By taking the log base \(n\) of both sides of the equation, we arrive at an exponential form, namely \(b^y=M\). It follows that
\[\begin{align*} {\log}_n(b^y)&= {\log}_nM \qquad \text{Apply the one-to-one property}\\[4pt] y{\log}_nb&= {\log}_nM \qquad \text{Apply the power rule for logarithms}\\[4pt] y&= \dfrac{{\log}_nM}{{\log}_nb} \qquad \text{Isolate y}\\[4pt] {\log}_bM&= \dfrac{{\log}_nM}{{\log}_nb} \qquad \text{Substitute for y} \end{align*}\]
For example, to evaluate \({\log}_536\) using a calculator, we must first rewrite the expression as a quotient of common or natural logs. We will use the common log.
\[\begin{align*} {\log}_536&= \dfrac{\log(36)}{\log(5)} \qquad \text{Apply the change of base formula using base 10}\\[4pt] &\approx 2.2266 \qquad \text{Use a calculator to evaluate to 4 decimal places} \end{align*}\]
The change-of-base formula can be used to evaluate a logarithm with any base.
For any positive real numbers \(M\), \(b\), and \(n\), where \(n≠1\) and \(b≠1\),
\[{\log}_bM=\dfrac{{\log}_nM}{{\log}_nb}\]
It follows that the change-of-base formula can be used to rewrite a logarithm with any base as the quotient of common or natural logs.
\[{\log}_bM=\dfrac{\ln M}{\ln b}\]
and
\[{\log}_bM=\dfrac{\log M}{\log b}\]
- Determine the new base \(n\), remembering that the common log, \(\log(x)\), has base 10, and the natural log, \(\ln(x)\),has base \(e\).
-
Rewrite the log as a quotient using the change-of-base formula
- The numerator of the quotient will be a logarithm with base \(n\) and argument \(M\).
- The denominator of the quotient will be a logarithm with base \(n\) and argument \(b\).
Change \({\log}_53\) to a quotient of natural logarithms.
Solution
Because we will be expressing \({\log}_53\) as a quotient of natural logarithms, the new base, \(n=e\).
We rewrite the log as a quotient using the change-of-base formula. The numerator of the quotient will be the natural log with argument \(3\). The denominator of the quotient will be the natural log with argument 5.
\({\log}_bM=\dfrac{\ln M}{\ln b}\)
\({\log}_53=\dfrac{\ln3}{\ln5}\)
Change \(\log0.58\) to a quotient of natural logarithms.
- Answer
-
\(\dfrac{\ln8}{\ln 0.5}\)
Yes. Remember that \(\log9\) meansl \({\log}_{10}9\). So, \(\log9=\dfrac{\ln9}{\ln10}\).
Evaluate \({\log}_2(10)\) using the change-of-base formula with a calculator.
Solution
According to the change-of-base formula, we can rewrite the log base \(2\) as a logarithm of any other base. Since our calculators can evaluate the natural log, we might choose to use the natural logarithm, which is the log base \(e\).
\[\begin{align*} {\log}_210&= \dfrac{\ln10}{\ln2} \qquad \text{Apply the change of base formula using base } e\\[4pt] &\approx 3.3219 \qquad \text{Use a calculator to evaluate to 4 decimal places} \end{align*}\]
Evaluate \({\log}_5(100)\) using the change-of-base formula.
- Answer
-
\(\dfrac{\ln100}{\ln5}≈\dfrac{4.6051}{1.6094}=2.861\)
Access these online resources for additional instruction and practice with laws of logarithms.
- The Properties of Logarithms
- Expand Logarithmic Expressions
- Evaluate a Natural Logarithmic Expression
Key Equations
| The Product Rule for Logarithms | \({\log}_b(MN)={\log}_b(M)+{\log}_b(N)\) |
| The Quotient Rule for Logarithms | \({\log}_b(\dfrac{M}{N})={\log}_bM−{\log}_bN\) |
| The Power Rule for Logarithms | \({\log}_b(M^n)=n{\log}_bM\) |
| The Change-of-Base Formula | \({\log}_bM=\dfrac{{\log}_nM}{{\log}_nb}\) \(n>0\), \(n≠1\), \(b≠1\) |
Key Concepts
- We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms. See Example \(\PageIndex{1}\).
- We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms. See Example \(\PageIndex{2}\).
- We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the log of its base. See Example \(\PageIndex{3}\) , Example \(\PageIndex{4}\), and Example \(\PageIndex{5}\).
- We can use the product rule, the quotient rule, and the power rule together to combine or expand a logarithm with a complex input. See Example \(\PageIndex{6}\) , Example \(\PageIndex{7}\) , and Example \(\PageIndex{8}\).
- The rules of logarithms can also be used to condense sums, differences, and products with the same base as a single logarithm. See Example \(\PageIndex{9}\) , Example \(\PageIndex{10}\), Example \(\PageIndex{11}\), and Example \(\PageIndex{12}\).
- We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of-base formula. See Example \(\PageIndex{13}\).
- The change-of-base formula is often used to rewrite a logarithm with a base other than 10 and \(e\) as the quotient of natural or common logs. That way a calculator can be used to evaluate. See Example \(\PageIndex{14}\).


1.1 Linguistics is Science
1. What does it mean to say that Linguistics is a science?
The field uses empirical observations to develop theories of language behaviour.
2. Each of the following sentences represents something someone might say about language. Which of them illustrates a
descriptive
view of language?
The use of quotative
like
in sentences such as, “She was like, I can’t believe you did that!” began to enter Canadian English with the generation of speakers born in 1971.
3. Which of the following kinds of data would a linguist be likely to observe?
Whether Korean includes tones that change the meaning of words.
1.2 Mental Grammar
1. Newspaper headlines occasionally have unexpectedly funny interpretations. One example is: T
wo cars were reported stolen by the police yesterday.
Which part of your mental grammar leads to the possibility that the police could have done the stealing or the reporting in this headline?
Syntax, Semantics.
2. Newfoundland English has some characteristic differences to standard Canadian English. The following sentences are grammatical in Newfoundland English:
I eats toast for breakfast every day
.
You knows the answer to that question.
What part of the mental grammar of Newfoundland English is different to Canadian English in these examples?
Morphology.
3. When speakers of Hawaiian pronounce the English phrase, “Merry Christmas”, it sounds like:
mele kalikimaka
. What part of the mental grammar of Hawaiian is responsible for how the English phrase gets pronounced?
Phonology.
1.3 Creativity & Generativity
1. What does it mean to say that mental grammar is generative?
The principles of mental grammar allow us to form completely novel sentences, and to understand them when we hear them.
2. The systematic principles of English phonology generate some word forms but not others. Which of the following words could be a possible word in English?
Klaff.
3. The systematic principles of English syntax generate some sentences but not others. Which of the following sentences is not possible in English?
Herself have wrote these excellent book.
1.4 Fundamental Properties of Language
1. It’s important to study Latin because Latin is more logical than other languages.
False.
2.
False.
3. The dictionary gives the only correct meaning and pronunciation for words.
False.
2.1 How Humans Produce Speech
1. What is the voicing of the
last
sound in the word ‘soup’?
Voiceless.
2. What is the voicing of the
last
sound in the word ‘life’?
Voiceless.
3. What is the voicing of the
last
sound in the word ‘seem’?
Voiced.
2.2 Articulators
1. What is the place of articulation of the FIRST sound in the word ‘minor’?
Bilabial.
2. What is the place of articulation of the FINAL sound in the word ‘wit’?
Alveolar.
3. What is the place of articulation of the FIRST sound in the word ‘photography’?
Labiodental.
2.3 Describing Speech Sounds: the IPA
1. The vowel sounds in the words
neat
and
spread
are both spelled “ea”. Do the vowels in the two words sound the same as each other or different?
Different.
2. Are the final sounds in the words
face
and
mess
the same as each other or different?
Same.
3. Are the first sounds in the two words
gym
and
gum
the same as each other or different?
Different.
2.5 Sonority, Consonants & Vowels
1. What kind of sound is the first sound in the word
early
?
Vowel.
2. What kind of sound is the first sound in the word
junior
?
Voiced consonant.
3. What kind of sound is the first sound in the word
winter
?
Glide.
2.6 Classifying Consonants
1. What is the articulatory description for the consonant sound represented by the IPA symbol [p]?
Voiceless bilabial stop.
2. What is the correct articulatory description for the consonant sound represented by the IPA symbol [ð]?
Voiced dental fricative.
3. What is the correct articulatory description for the consonant sound represented by the IPA symbol [ʃ]?
Voiceless post-alveolar fricative.
2.7 Classifying Vowels
1. What is the articulatory description for the vowel [i]?
High front unrounded tense vowel.
2. What is the articulatory description for the vowel [ɛ]?
Mid front unrounded lax vowel.
3. What is the articulatory description for the vowel [ɑ]?
Low back unrounded tense vowel.
2.8 Diphthongs
1. What is the diphthong sound in the word in the word
proud
?
[aʊ].
2. What is the diphthong sound in the word in the word
rain
?
[eɪ].
3. What is the diphthong sound in the word in the word
sigh
?
[aɪ].
3.3 Syllabic Consonants
1. The video indicated that the word
funnel
can be transcribed to indicate that the second syllable consists of a syllabic [l̩]. The word
elbow
is also spelled with the letters ‘el’. Say the two words to yourself several times. Which is the correct transcription for
elbow
?
[ɛlboʊ].
2. The words
human
and
manager
both contain a syllable that is spelled with the letters ‘man’. In which word does that syllable contain a syllabic [n̩]?
Human.
3. In the word
umbrella
, is the [m] syllabic?
No.
3.4 Aspirated Stops in English
1. The following city names all contain the letter ‘t’ within the word. In which of them is the letter ‘t’ pronounced as [
tʰ
]?
Victoria.
2. The following words all contain the segment /k/. In which of them is it pronounced as the allophone [kʰ]?
Accomplish.
3. The following words all contain the segment /p/. In which of them is it pronounced as the allophone [pʰ]?
Appearance.
3.5 Articulatory Processes: Assimilation
1. What articulatory process is at work when the word
bank
is pronounced as [bæŋk]?
Assimilation (Anticipatory / Regressive).
2. What articulatory process is at work when a child pronounces the word
yellow
as [lɛloʊ]?
Assimilation (Anticipatory / Regressive).
3. What articulatory process is at work when the word
cream
is pronounced as [kɹ̥ijm]?
Assimilation (Perseveratory / Progressive).
3.6 Other Articulatory Processes
1. What articulatory process is at work when the word
idea
is pronounced as [ajdijɚ]?
Epenthesis.
2. What articulatory process is at work when the word
gorilla
is pronounced as [ɡɹɪlʌ]?
Deletion.
3. What articulatory process is at work when the word
you
is pronounced as [jə]?
Reduction.
3.7 Suprasegmentals
1. Young children’s voices are usually recognizably different from adult’s voices. Which factor is likeliest to be different between children’s speech and adults’ speech?
Pitch.
2. In English, yes-no questions often conclude with rising pitch, whereas wh-questions often have a falling pitch on the final words. Is this pitch difference a difference in tone or in intonation?
Intonation.
3. English uses pitch as one factor in syllable stress. There are many English pairs of words like
record
(noun) and
record
(verb), which are spelled the same but differ in their stress patterns. Which of the following is true for this pair of words?
The first syllable has a higher pitch than the second in the noun record.
4.1 Phonemes and Contrast
1. Are the phonetically different segments [m] and [n] phonemically contrastive in English?
Yes.
2. Are the phonetically different segments [p] and [pʰ] phonemically contrastive in English?
No.
3. Do the words
sight
and
site
form a minimal pair in English?
No.
4.2 Allophones and Predictable Variation
1. Remember that in English, voiceless stops are aspirated at the beginning of a word and the beginning of a stressed syllable, but never in the middle of a word nor at the end of a word. Which term best describes this pattern?
Complementary distribution.
2. The symbol [l̴] represents a velarized [l]. Looking at the following set of transcribed English words, what can you conclude about [l] and [l̴] in English?
[l] and [l̴] are in complementary distribution in English.
leaf [lif] fall [fɑl̴]
luck [lʌk] spill [spɪl̴]
lemon [lɛmən] wolf [wʊl̴f]
3. Remembering that the alveolar flap [ɾ] appears in a predictable environment in English (see Section 3.9), which statement is true for English?
The segments [t] and [ɾ] are allophones of the same phoneme in English.
4.3 Phonetic Segments and Features
1. These two segments — [w] [o] — have many features in common. Which feature distinguishes them?
[syllabic].
2. These two segments — [p] [f] — have many features in common. Which feature distinguishes them?
[continuant].
3. These two segments — [p] [b] — have many features in common. Which feature distinguishes them?
[voice].
4.4 Natural Classes
1. In the following set of segments, which segment must be excluded to make the remaining segments constitute a natural class?
[æ].
2. In the following set of segments, which segment must be excluded to make the remaining segments constitute a natural class?
[f].
3. This set of segments constitutes a natural class: [i ɛ æ]. Which segment could you add to the set while still preserving the natural class?
[ɪ].
4.5 Phonological Derivations
1. Which phonological rule accurately represents the process, “vowels become nasalized before a nasal consonant”?
2. Which sentence accurately describes the process depicted in this phonological rule? Voiceless fricatives become voiced between voiced sonorants
3. Which sentence accurately describes the process depicted in this phonological rule? The segment [ə] is epenthesized following a strident consonant.
5.1 How Babies Learn the phoneme categories of their language
1. The phonology of Thai contains a three-way contrast between voiced /b/, voiceless /p/, and aspirated /pʰ/. How likely is it that a six-month-old baby growing up in an English-speaking household would be able to hear the difference between Thai [p] and [pʰ]?
Very likely.
2. Arabic phonology includes a contrast between uvular and pharyngeal fricatives. How likely is it that a two-year-old growing up in an English-speaking household would be able to hear the difference between these two places of articulation?
Very unlikely.
3. In the various dialects of Chinese, there is no phonemic contrast between tense and lax vowels. How likely is it that a four-month-old baby growing up in a Mandarin Chinese-speaking household would be able to hear the contrast between English [e] and [ɛ]?
Very likely.
5.2 How Adults Learn
1. In Canadian French, the front tense vowels [y] and [i] become the lax vowels [ʏ] and [ɪ] in certain environments. Is it likely to be easy or difficult for a native speaker of Canadian French to learn the English contrast between /i/ and /ɪ/ (as in
sleep
and
slip
)?
Difficult.
2. Arabic includes a contrast between a voiceless velar fricative [x] and a voiced velar fricative [ɣ]. Is this contrast likely to be easy or difficult for native speakers of English to learn?
Easy.
3. In Cree, voiceless stops become voiced between vowels. Given that Cree has both voiced and voiceless stops in its phonetic inventory, is it likely to be easy or difficult for a native speaker of Cree to learn the phonemic contrast between /b/ and /p/ in English?
Difficult.
6.3 Inflectional Morphology
1. What type of grammatical information does the inflectional affix in the word
speeches
communicate?
Number.
2. What type of grammatical information does the inflectional affix in the word
climbed
communicate?
Tense.
3. What type of grammatical information does the inflectional difference between
he
and
him
indicate?
Case.
6.4 Derivational Morphology
1. Which of the following best describes the derivation of the word
assignment
?
Verb + -ment ➔ Noun.
2. Which of the following best describes the derivation of the word
skillful
?
Noun + -ful ➔ Adjective.
3. Which of the following best describes the derivation of the word
simplify
?
Adjective + -ify ➔ Verb.
7.2 Compound Words
1. In the sentence, “The room contained a bearskin rug,” what kind of compound is
bearskin
?
Endocentric.
2. In the sentence, “Randy worked as a cowhand on the ranch,” what kind of compound is
cowhand
?
Exocentric.
3. In the sentence, “Hyunji manages a bookshop,”, what kind of compound is
bookshop
?
Endocentric.
7.5 Neurolinguistics: Syntactic Category Differences in the Brain
1. Comparing the following sets of words, which would you predict would lead to greater blood flow in more areas of the brain?
Humming, singing, whistling.
2. When shown a picture of a pair of tongs, a patient describes the picture, “You pick up things with it”. Which type of aphasia is this response more typical of?
Anomic aphasia.
3. When describing an injury to his knees, a patient says, “no good uh ache and uh uh uh knees and ankles uh home doctor and legs”. Which type of aphasia is this response more typical of?
Agrammatic aphasia.
8.1 Tree Diagrams
In the following tree diagram:
1. What is the structural relationship between V
loves
and NP
sushi
?
V and NP are sisters.
2. What is the structural relationship between NP
Colin
and V
loves
?
NP and V are not related in any of these three ways.
3. Which node is the sister of NP
Colin
?
T’
8.2 X-bar Phrase Structure
1. In this tree diagram, what position does the DP
my coffee
occupy?
Complement.
2. In this tree diagram, what position does the D
my
occupy?
Head.
3. In this tree diagram, what position does the NP
Hamilton
occupy?
Complement.
8.4 Sentences are Phrases
1. Which of the following is the correct representation for the sentence
Sara bought a car
?
2. Which of the following is the correct representation for the sentence
Sang-Ho won a medal
?
3. Which of the following is the correct representation for the sentence
Prabhjot should read this article
?
8.6 Subcategories
1. What is the subcategory of the underlined verb in this sentence? The soccer players
kicked
the ball.
Transitive.
2. What is the subcategory of the underlined verb in this sentence? Many birds
fly
over Ontario each fall.
Intransitive.
3. What is the subcategory of the underlined verb in this sentence? This game
teaches
children the alphabet.
Ditransitive.
8.8 Adjuncts
1. Is the underlined phrase an adjunct or a complement?
Sam ran
the Around-the-Bay race
.
Complement.
2. Is the underlined phrase an adjunct or a complement?
Sam ran
this morning
.
Adjunct.
3. Is the underlined phrase an adjunct or a complement?
The baby slept
through the night
.
Adjunct.
8.9 Move
1. Which tree diagram correctly represents the question,
“Could you hand me those scissors?”
2. Which tree diagram correctly represents the question,
“Does Suresh like Ethiopian food?”
3. Which tree diagram correctly represents the question,
“Will the Habs win the Stanley Cup?”
8.10 Wh-Movement
1. Which tree diagram correctly represents the Deep Structure for the question,
“Who did Brenda see at the gym?”
2.Which tree diagram correctly represents the Surface Structure for the question,
“Where did you get that hat?”
3. Which tree diagram correctly represents the Surface Structure for the question,
“Why should I trust you?”
8.12 Psycholinguistics: Traces in the Mind
1. Which of the following illustrates the position of the trace in the wh-question W
hat did Christina order at Chipotle?
What did Christina order what at Chipotle.
2. Which of the following ungrammatical sentences gives evidence that unpronounced traces exist in our mental representations of sentences?
*What did you eat sandwiches for lunch?
3. Predict which sentence would lead to more eye movements to a picture of a rabbit after the verb
chase
.
What did the fox chase ^ into the hedge?
9.2 Events, Participants, and Thematic Roles
1. What label best describes the thematic role of the underlined NP?
The guard
chased the intruder.
Agent.
2. What label best describes the thematic role of the underlined NP?
The wind
slammed the door shut.
Cause.
3. What label best describes the thematic role of the underlined NP? The guard followed
the intruder
.
Theme.
9.3 Thematic Roles and Passive Sentences
1. Is the following sentence in the active or passive voice? The patient was diagnosed with alopecia.
Passive.
2. Is the following sentence in the active or passive voice? Eileen was convinced that her appointment had been cancelled.
Passive.
3. Is the following sentence in the active or passive voice? The children had been invited to a tea party.
Passive.
9.4 Neurolinguistics: Using EEG to Investigate Syntax and Semantics
1. What kind of ERP is likely to be observed in response to this sentence? The children’s feet have grown so I bought them some new
spoons
.
N400.
2. What kind of ERP is likely to be observed in response to this sentence? Elias told the coach that he wanted to learn to
swam
.
P600.
3. What kind of ERP is likely to be observed in response to this sentence? Before the dinner party, Isla spent a long time in the
cooked
.
P600.
9.5 Neurolinguistics and Second Language Learning
1. French uses morphology indicate whether nouns, adjectives and determiners are masculine or feminine. If an L1 speaker of English is learning French, what kind of transfer are they likely to experience in learning this property of French grammar?
Negative transfer.
2. Russian does not have definite or indefinite determiners like English
a
and
the
. If an L1 speaker of Russian is learning English, what kind of transfer are they likely to experience in learning this property of English grammar?
Negative transfer.
3. Russian groups nouns by their grammatical gender, either masculine, feminine or neuter. Look again at the facts about French presented in Question 1. If an L1 speaker of Russian is learning French, what kind of transfer are they likely to experience in learning this property of French grammar?
Positive transfer.
10.1 Elements of Word Meaning: Intensions and Extensions
1. Sidney Crosby, Wayne Gretzy, and Maurice Richard, and ______________ of
hockey player
.
Extensions.
2.
Has pages, binding
, and
contains writing
are all __________________ of
book
.
Intensions.
3.
Head of state, lives in the White House
, and
American
are all ______________ of
President of the United States
.
Intensions.
10.2 Intensions in the Mind
1. For most speakers of Canadian English in the category
animals, giraffe
is probably:
Less typical than do
g.
2. For most speakers of Canadian English, in the category
pets, tarantula
is probably:
Peripheral
.
3. Choose the set of features that best defines
chair
.
[+furniture, +legs, +back, +seat, -blankets]
.
10.3 Psycholinguistics of Word Meanings
1. Which of the following words would we expect to prime a target word
carrot
?
Broccoli.
2. Which of the following words would we expect to prime a target word
happy
?
Sad.
3. Which of the following words would we expect to prime a target word
week
?
Month.


27.2A: Fertilization
Fertilization occurs when a sperm and an egg have fused together to form a zygote, which begins to divide as it moves towards the uterus.
- Describe human fertilization
Key Points
- Fertilization is commonly known as conception. Once the fertilized gamete (ovum) implants itself in the uterine lining, pregnancy begins.
- The fusion of male and female gametes ( sperm and ovum, respectively) usually occurs following the act of sexual intercourse. However, artificial insemination and in vitro fertilization have made achieving pregnancy possible without engaging in sexual intercourse.
- The process of fertilization occurs in several steps and the interruption of any of them can lead to failure.
- Prior to fertilization, sperm undergo a process of capacitation in response to conditions in the female reproductive tract, which include increases in motility and destabilization of the cell membrane that allows the head of the sperm to penetrate the egg.
Key Terms
- fertilization : The act of fecundating or impregnating animal or vegetable gametes.
- capacitation : A step spermatozoa undergo in the female reproductive track that renders them capable of fertilizing an oocyte.
- implantation : The embedding of the fertilized ovum into the uterine wall.
- Nondisjunction : Nondisjunction is the failure of homologous chromosomes or sister chromatids to separate properly during cell division.
- zygote : A fertilized egg cell.
If pregnancy is considered to begin at the point of implantation, the process leading to pregnancy occurs earlier as the result of the female gamete, or oocyte, merging with the male gamete, or spermatozoon. In medicine, this process is referred to as fertilization; in lay terms, it is more commonly known as conception.
After the point of fertilization the fused product of the female and male gamete is referred to as a zygote or fertilized egg.
However, the advent of artificial insemination and in vitro fertilization have made achieving pregnancy possible without engaging in sexual intercourse. This approach may be undertaken as a voluntary choice or due to infertility.
Human fertilization : The sperm and ovum unite through fertilization, creating a zygote that (over the course of 8–9 days) will implant in the uterine wall, where it will reside over the course of 9 months.
The process of fertilization occurs in several steps and the interruption of any of them can lead to failure. At the beginning of the process, the sperm undergoes a series of changes, as freshly ejaculated sperm is unable or poorly able to fertilize.
The sperm must undergo capacitation in the female’s reproductive tract over several hours, which increases its motility and destabilizes its membrane. By destabilizing the membrane, the sperm prepares for the acrosome reaction, the enzymatic penetration of the egg’s tough membrane, the zona pellucida. The sperm and the egg cell (which has been released from one of the female’s two ovaries) unite in one of the two fallopian tubes.
The fertilized egg, known as a zygote, then moves toward the uterus, a journey that can take up to a week to complete until implantation occurs. Through fertilization, the egg is activated to begin its developmental process (progressing through meiosis II), and the haploid nuclei of the two gametes come together to form the genome of a new diploid organism.
Nondisjunction during the completion of meiosis or problems with early cell division in the zygote to blastula stages can lead to problems with implantation and pregnancy failure.


4.6: Applications
The lengthy derivation in Secs. 4.3–4.5 is based on the Kelvin–Planck statement describing the impossibility of converting completely into work the energy transferred into the system by heat from a single heat reservoir. The derivation has now given us all parts of the mathematical statement of the second law shown in the box in Sec. 4.2. The mathematical statement includes an equality, \(\dif S=\dq/T\bd\), that applies to an infinitesimal reversible change, and an inequality, \(\dif S > \dq/T\bd\), that applies to an infinitesimal irreversible change. It is convenient to combine the equality and inequality in a single relation that is a general mathematical statement of the second law: \begin{gather} \s{\dif S \geq \frac{\dq}{T\bd}} \tag{4.6.1} \cond{(\({}\subs{ rev}\sups{ irrev}\), closed system)} \end{gather} The inequality refers to an irreversible change and the equality to a reversible change, as indicated by the notation \({}\subs{ rev}\sups{ irrev}\) in the conditions of validity. The integrated form of this relation is \begin{gather} \s{\Del S \geq \int\!\frac{\dq}{T\bd}} \tag{4.6.2} \cond{(\({}\subs{ rev}\sups{ irrev}\), closed system)} \end{gather}
During a reversible process, the states are equilibrium states and the temperature is usually uniform throughout the system. The only exception is if the system happens to have internal adiabatic partitions that allow phases of different temperatures in an equilibrium state. When the process is reversible and the temperature is uniform, we can replace \(\dif S=\dq/T\bd\) by \(\dif S=\dq/T\).
The rest of Sec. 4.6 will apply Eqs. 4.6.1 and 4.6.2 to various reversible and irreversible processes.
4.6.1 Reversible heating
The definition of the heat capacity \(C\) of a closed system is given by Eq. 3.1.9: \(C \defn \dq/\dif T\). For reversible heating or cooling of a homogeneous phase, \(\dq\) is equal to \(T\dif S\) and we can write \begin{equation} \Del S = \int_{T_1}^{T_2}\!\frac{C}{T}\dif T \tag{4.6.3} \end{equation} where \(C\) should be replaced by \(C_V\) if the volume is constant, or by \(C_p\) if the pressure is constant (Sec. 3.1.5). If the heat capacity has a constant value over the temperature range from \(T_1\) to \(T_2\), the equation becomes \begin{equation} \Del S = C\ln\frac{T_2}{T_1} \tag{4.6.4} \end{equation} Heating increases the entropy, and cooling decreases it.
4.6.2 Reversible expansion of an ideal gas
When the volume of an ideal gas, or of any other fluid, is changed reversibly and adiabatically , there is of course no entropy change.
When the volume of an ideal gas is changed reversibly and isothermally , there is expansion work given by \(w=-nRT\ln(V_2/V_1)\) (Eq. 3.5.1). Since the internal energy of an ideal gas is constant at constant temperature, there must be heat of equal magnitude and opposite sign: \(q=nRT\ln(V_2/V_1)\). The entropy change is therefore \begin{gather} \s{\Del S = nR\ln\frac{V_2}{V_1}} \tag{4.6.5} \cond{(reversible isothermal volume} \nextcond{change of an ideal gas)} \end{gather} Isothermal expansion increases the entropy, and isothermal compression decreases it.
Since the change of a state function depends only on the initial and final states, Eq. 4.6.5 gives a valid expression for \(\Del S\) of an ideal gas under the less stringent condition \(T_2=T_1\); it is not necessary for the intermediate states to be equilibrium states of the same temperature.
4.6.3 Spontaneous changes in an isolated system
An isolated system is one that exchanges no matter or energy with its surroundings. Any change of state of an isolated system that actually occurs is spontaneous, and arises solely from conditions within the system, uninfluenced by changes in the surroundings—the process occurs by itself, of its own accord. The initial state and the intermediate states of the process must be nonequilibrium states, because by definition an equilibrium state would not change over time in the isolated system.
Unless the spontaneous change is purely mechanical, it is irreversible. According to the second law, during an infinitesimal change that is irreversible and adiabatic, the entropy increases. For the isolated system, we can therefore write \begin{gather} \s{\dif S > 0} \tag{4.6.6} \cond{(irreversible change, isolated system)} \end{gather} In later chapters, the inequality of Eq. 4.6.6 will turn out to be one of the most useful for deriving conditions for spontaneity and equilibrium in chemical systems: The entropy of an isolated system continuously increases during a spontaneous, irreversible process until it reaches a maximum value at equilibrium .
If we treat the universe as an isolated system (although cosmology provides no assurance that this is a valid concept), we can say that as spontaneous changes occur in the universe, its entropy continuously increases. Clausius summarized the first and second laws in a famous statement: Die Energie der Welt ist constant; die Entropie der Welt strebt einem Maximum zu (the energy of the universe is constant; the entropy of the universe strives toward a maximum).
4.6.4 Internal heat flow in an isolated system
Suppose the system is a solid body whose temperature initially is nonuniform. Provided there are no internal adiabatic partitions, the initial state is a nonequilibrium state lacking internal thermal equilibrium. If the system is surrounded by thermal insulation, and volume changes are negligible, this is an isolated system. There will be a spontaneous, irreversible internal redistribution of thermal energy that eventually brings the system to a final equilibrium state of uniform temperature.
In order to be able to specify internal temperatures at any instant, we treat the system as an assembly of phases, each having a uniform temperature that can vary with time. To describe a region that has a continuous temperature gradient, we approximate the region with a very large number of very small phases or parcels, each having a temperature infinitesimally different from its neighbors.
We use Greek letters to label the phases. The temperature of phase \(\pha\) at any given instant is \(T\aph\). We can treat each phase as a subsystem with a boundary across which there can be energy transfer in the form of heat. Let \(\dq_{\pha\phb}\) represent an infinitesimal quantity of heat transferred during an infinitesimal interval of time to phase \(\pha\) from phase \(\phb\). The heat transfer, if any, is to the cooler from the warmer phase. If phases \(\pha\) and \(\phb\) are in thermal contact and \(T\aph\) is less than \(T\bph\), then \(\dq_{\pha\phb}\) is positive; if the phases are in thermal contact and \(T\aph\) is greater than \(T\bph\), \(\dq_{\pha\phb}\) is negative; and if neither of these conditions is satisfied, \(\dq_{\pha\phb}\) is zero.
To evaluate the entropy change, we need a reversible path from the initial to the final state. The net quantity of heat transferred to phase \(\pha\) during an infinitesimal time interval is \(\dq\aph = \sum_{\phb\neq\pha}\dq_{\pha\phb}\). The entropy change of phase \(\pha\) is the same as it would be for the reversible transfer of this heat from a heat reservoir of temperature \(T\aph\): \(\dif S\aph=\dq\aph/T\aph\). The entropy change of the entire system along the reversible path is found by summing over all phases: \begin{equation} \begin{split} \dif S & = \sum_\pha\dif S\aph = \sum_\pha \frac{\dq\aph}{T\aph} = \sum_\pha\sum_{\phb\neq\pha}\frac{\dq_{\pha\phb}}{T\aph}\ & = \sum_\pha\sum_{\phb > \pha} \left(\frac{\dq_{\pha\phb}}{T\aph} +\frac{\dq_{\phb\pha}}{T\bph}\right) \end{split} \tag{4.6.7} \end{equation} There is also the condition of quantitative energy transfer, \(\dq_{\phb\pha}=-\dq_{\pha\phb}\), which we use to rewrite Eq. 4.6.7 in the form \begin{equation} \dif S = \sum_\pha\sum_{\phb > \pha} \left(\frac{1}{T\aph}-\frac{1}{T\bph} \right)\dq_{\pha\phb} \tag{4.6.8} \end{equation}
Consider an individual term of the sum on the right side of Eq. 4.6.8 that has a nonzero value of \(\dq_{\pha\phb}\) due to finite heat transfer between phases \(\pha\) and \(\phb\). If \(T\aph\) is less than \(T\bph\), then both \(\dq_{\pha\phb}\) and \((1/T\aph-1/T\bph)\) are positive. If, on the other hand, \(T\aph\) is greater than \(T\bph\), both \(\dq_{\pha\phb}\) and \((1/T\aph-1/T\bph)\) are negative. Thus each term of the sum is either zero or positive, and as long as phases of different temperature are present, \(\dif S\) is positive.
This derivation shows that during a spontaneous thermal equilibration process in an isolated system, starting with any initial distribution of the internal temperatures, the entropy continuously increases until the system reaches a state of thermal equilibrium with a single uniform temperature throughout. The result agrees with Eq. 4.6.6. Harvey S. Leff ( Am. J. Phys. , 45 , 252–254, 1977) obtains the same result by a more complicated derivation.
4.6.5 Free expansion of a gas
Consider the free expansion of a gas shown in Fig. 3.8. The system is the gas. Assume that the vessel walls are rigid and adiabatic, so that the system is isolated. When the stopcock between the two vessels is opened, the gas expands irreversibly into the vacuum without heat or work and at constant internal energy. To carry out the same change of state reversibly, we confine the gas at its initial volume and temperature in a cylinder-and-piston device and use the piston to expand the gas adiabatically with negative work. Positive heat is then needed to return the internal energy reversibly to its initial value. Because the reversible path has positive heat, the entropy change is positive.
This is an example of an irreversible process in an isolated system for which a reversible path between the initial and final states has both heat and work.
4.6.6 Adiabatic process with work
In general (Sec. 3.10), an adiabatic process with a given initial equilibrium state and a given change of a work coordinate has the least positive or most negative work in the reversible limit. Consider an irreversible adiabatic process with work \(w\subs{irr}\). The same change of state can be accomplished reversibly by the following two steps: (1) a reversible adiabatic change of the work coordinate with work \(w\rev\), followed by (2) reversible transfer of heat \(q\rev\) with no further change of the work coordinate. Since \(w\rev\) is algebraically less than \(w\subs{irr}\), \(q\rev\) must be positive in order to make \(\Del U\) the same in the irreversible and reversible paths. The positive heat increases the entropy along the reversible path, and consequently the irreversible adiabatic process has a positive entropy change. This conclusion agrees with the second-law inequality of Eq. 4.6.1.


9.13: Reading- Brand Development Strategies
Managing Brands As Strategic Assets
As organizations establish and build strong brands, they can pursue a number of strategies to continue developing them and extending their value to stakeholders (customers, retailers, supply chain and distribution partners, and of course the organization itself).
Brand Ownership
Who “owns” the brand? The legal owner of a brand is generally the individual or entity in whose name the legal registration has been filed. Operationally speaking, brand ownership should be the responsibility of an organization’s management and employees. Brand ownership is about building and maintaining a brand that reflects your principles and values. Brand building is about effectively persuading customers to believe in and purchase your product or service. Iconic brands, such as Apple and Disney, often have a history of visionary leaders who champion the brand, evangelize about it, and build it into the organizational culture and operations.
When an organization truly owns its brand, its efforts are unified around a common symbol of the value it provides to customers. These organizations use their resources wisely to produce marketing that is targeted and effective because they have a sophisticated understanding of the marketplace; they know how their brand and offerings fit into it, which audiences they are targeting, and they have a strategy for successful growth. These advantages lead to disciplined and effective brand management, which enables these organizations to remain relevant in a rapidly changing and often saturated marketplace.
Branding Strategies
A branding strategy helps establish a product within the market and to build a brand that will grow and mature. Making smart branding decisions up front is crucial since a company may have to live with their decisions for a long time. The following are commonly used branding strategies:
“Branded House” Strategy
A “branded house” strategy (sometimes called a “house brand”) uses a a strong brand—typically the company name—as the identifying brand name for a range of products (for example, Mercedes Benz or Black & Decker) or a range of subsidiary brands (such as Cadbury Dairy Milk or Cadbury Fingers). Because the primary focus and investment is in a single, dominant “house” brand, this approach can be simpler and more cost effective in the long run when it is well aligned with broader corporate strategy.
“House of Brands” Strategy
With the “house of brands” strategy, a company invests in building out a variety of individual, product-level brands. Each of these brands has a separate name and may not be associated with the parent company name at all. These brands may even be in de facto competition with other brands from the same company. For example, Kool-Aid and Tang are two powdered beverage products, both owned by Kraft Foods. The “house of brands” strategy is well suited to companies that operate across many product categories at the same time. It allows greater flexibility to introduce a variety of different products, of differing quality, to be sold without confusing the consumer’s perception of what business the company is in or diluting brand perceptions about products that target different tiers or types of consumers within the same product category.
Competitive Multi-Brand Strategy
In a very saturated market, a supplier can deliberately launch totally new brands in apparent competition with its own existing strong brand (and often with identical product characteristics) to soak up some of the share of the market. The rationale is that having three out of twelve brands in such a market will give a greater overall share than having one out of ten. Procter & Gamble is a leading exponent of this philosophy, running as many as ten detergent brands in the U.S. market. In 2015, hotel giant Marriott International operated sixteen different hotel chains across different pricing tiers, including some chains that compete with one another directly. A sampling of these includes Fairfield Inn, Springhill Suites, Residence Inn, Courtyard, Marriott, JW Marriott, and The Ritz Carlton, among others.
Cannibalization is a particular problem with the multi-brands-strategy. As will be discussed further in the product marketing module, cannibalization occurs when the new brand takes business away from an established one, which the organization also owns. This may be acceptable (indeed expected) if there is a net gain overall.
Brand Families, or “Umbrella Branding”
Similar to a “branded house” strategy, a brand family uses a single brand name for multiple products. However, brand families–also called umbrella branding–may also be used in a “house of brands” strategy to extend the reach of some of the company’s brands. For instance, consumer products powerhouse Procter & Gamble manages many popular brands including Tide (laundry detergent), Pampers (disposable diapers), Ivory (soap), and Olay (skin care and beauty products) among many others. Each of these brands constitutes its own family, with multiple products carrying the same brand name.
Attitude Branding and Iconic Brands
Attitude branding is a strategy of representing the larger feeling that a brand comes to embody. The idea is that the brand’s feeling or “attitude” transcends the specific products being consumed. Examples of companies that use this approach effectively include:
- Nike: “Just do it”
- Apple: “Think different”
- Patagonia: “We’re in business to save the planet.”
Effective attitude branding can transform strong brands into iconic, “lifestyle” brands that contribute to the consumer’s self-expression and personal identity.
Component Branding
Some suppliers of important product or manufacturing components try to guarantee positions of preference by promoting these components as brands in their own right. For example, Intel created competitive advantage for itself in the PC market with the slogan (and famous sticker) “Intel Inside.”
Private-Label or Store Branding
Also called store branding, private-label branding has become increasingly popular. In cases where the retailer has a particularly strong identity, the private label may be able to compete against even the strongest brand leaders and may outperform those products that are not otherwise strongly branded. The northeastern U.S. grocery chain Wegman’s offers many grocery products that carry the Wegman’s brand name. Meanwhile national grocery chain Safeway offers several different private label “store” brands: Signature Select, O Organics, Signature Cafe, and Primo Taglio, among others. [1]
“No-Brand” Branding
A number of companies successfully pursue “no-brand” strategies by creating packaging that imitates generic-brand simplicity. “No brand” branding can be considered a type of branding since the product is made conspicuous by the absence of a brand name. “Tapa Amarilla” or “Yellow Cap” in Venezuela during the 1980s is a prime example of no-brand strategy. It was recognized simply by the color of the cap of this cleaning products company.
Personal and Organizational Brands
Personal and organizational branding are strategies for developing a brand image and marketing engine around individual people or groups. Personal branding treats persons and their careers as products to be branded and sold to target audiences. Organizational branding promotes the mission, goals, and/or work of the group being branded. The music and entertainment industries provide many examples of personal and organizational branding. From Justin Bieber to George Clooney to Kim Kardashian, virtually any celebrity today is a personal brand. Likewise, bands, orchestras, and other artistic groups typically cultivate an organizational (or group) brand. Faith branding is a variant of this brand strategy, which treats religious figures and organizations as brands seeking to increase their following. Mission-driven organizations such the Girl Scouts of America, the Sierra Club, the National Rifle Association (among millions of others) pursue organizational branding to expand their membership, resources, and impact.
Crowd-Sourced Branding
Crowd-sourced branding is the phenomenon of brands being created “by the people” for the business, which is the opposite of how branding traditionally works (business create the brands). This method minimizes the risk of brand failure, since the people who might reject the brand are the ones involved in the branding process. The drawback is that the business cannot fully control these brands, because they are the product of crowd sourcing and, in effect, are owned by “the crowd.”
An interesting example of crowd-sourced branding is the Timbers Army, the independent fan organization of the Portland Timbers Major League Soccer (MLS) Team. The Timbers Army was created by fans, and it operates independently from the MLS team and the Portland Timbers management. Although the organizations coordinate in many areas, ultimately the fan organization gets to assert and control its own brand identity.
Place Branding and Nation Branding
The developing fields of place branding and nation branding work on the assumption that places compete with other places to win over people, investment, tourism, economic development, and other resources. With this in mind, public administrators, civic leaders, and business groups may team up to “brand” and promote their city, region, or nation among target audiences. Depending on the goals they are trying to achieve, targets for these marketing initiatives may be real-estate developers, employers and business investors, tourists and tour/travel operators, and so forth. While place branding may focus on any given geographic area or destination, nation branding aims to measure, build, and manage the reputation of countries.
The city-state Singapore is an early, successful example of nation branding. The Las Vegas “What Happens Here, Stays Here” campaign, shown in in the following video, is a well-known example of place branding.
You can view the transcript for “What Happens Here, Stays Here – Sketchbook Commercial” (opens in new window) or the text alternative for “What Happens Here, Stays Here – Sketchbook Commercial” (opens in new window).
Co-Branding
Co-branding is an arrangement in which two established brands collaborate to offer a single product or service that carries both brand names. In these relationships, generally both parties contribute something of value to the new offering that neither would have been able to achieve independently. Effective co-branding builds on the complementary strengths of the existing brands. It can also allow each brand an entry point into markets in which they would not otherwise be credible players.
The following are some examples of co-branded offerings:
- Delta Airlines and American Express offer an entire family of co-branded credit cards; other airlines offer similar co-branded cards that offer customer rewards in terms of frequent flyer points and special offers.
- Home furnishings company Pottery Barn and the paint manufacturer Benjamin Moore co-brand seasonal color palettes for home interior paints
- Forever 21 worked with the USPS to create an exclusive line of clothing featuring USPS branding.
- Auto maker Fiat and toy maker Mattel teamed up to celebrate Barbie’s fiftieth anniversary with the nail-polish-pink Fiat 500 Barbie car.
Co-branding is a common brand-building strategy, but it can present difficulties. There is always risk around how well the market will receive new offerings, and sometimes, despite the best-laid plans, co-branded offerings fall flat. Also, these arrangements often involve complex legal agreements that are difficult to implement. Co-branding relationships may be unevenly matched, with the partners having different visions for their collaboration, placing different priority on the importance of the co-branded venture, or one partner holding significantly more power than the other in determining how they work together. Because co-branding impacts the existing brands, the partners may struggle with how to protect their current brands while introducing something new and possibly risky.
Brand Licensing
Brand licensing is the process of leasing or renting the right to use a brand in association with a product or set of products for a defined period and within a defined market, geography, or territory. Through a licensing agreement, a firm (licensor) provides some tangible or intangible asset to another firm (licensee) and grants that firm the right to use the licensor’s brand name and related brand assets in return for some payment. The licensee obtains a competitive advantage in this arrangement, while the licensor obtains inexpensive access to the market in question.
Licensing can be extremely lucrative for the owner of the brand, as other organizations pay for permission to produce products carrying a licensed name. The Walt Disney Company was an early pioneer in brand licensing, and it remains a leader in this area with its wildly popular entertainment and toy brands: Star Wars, Disney Princesses, Toy Story, Mickey Mouse, and so on. Toy manufacturers, for example, pay millions of dollars and vie for the rights to produce and sell products affiliated with these “super-brands.”
A licensing arrangement contains risk, in that if the licensing venture is very successful, the profit potential is limited by the terms of the licensing agreement. If the venture isn’t successful, the licensee loses a substantial investment, and the failure may reflect poorly on the original brand. Also, a licensor might be very controlling about how the licensed offering is designed, produced, distributed, marketed, or sold, making it difficult for the licensee to meet the expectations or requirements of the licensor. Conversely, a licensor might make a long-term commitment to a firm, and that firm could be less capable than expected, leading to a botched implementation of the licensing venture. Or, the licensee may be unwilling to invest in product quality, marketing, distribution, or other areas needed to be successful.
Franchising represents a very popular type of licensing arrangement for many consumer products firms. Holiday Inn, Hertz Car Rental, and McDonald’s have all expanded globally through franchising. In a franchise, the entity purchasing the franchise (the franchisee) typically pays an up-front fee plus a percentage of revenue in return for the right to use branded assets such as recognized brand name(s), proven products, building design and decor (as in a fast-food restaurant chain), business processes, and so forth.
Lines Extensions and Brand Extensions
Organizations use line extensions and brand extensions to leverage and increase brand equity.
A company creates a line extension when it introduces a new variety of offering within the same product category. To illustrate with the food industry, a company might add new flavors, package sizes, nutritional content, or products containing special additives in line extensions. Line extensions aim to provide more variety and hopefully capture more of the market within a given category. More than half of all new products introduced each year are line extensions. For example, M&M candy varieties such as peanut, pretzel, peanut butter, and dark chocolate are all line extensions of the M&M brand. Diet Coke™ is a line extension of the parent brand Coke ™. While the products have distinct differences, they are in the same product category.
A brand extension moves an existing brand name into a new product category, with a new or somehow modified product. In this scenario, a company uses the strength of an established product to launch a product in a different category, hoping the popularity of the original brand will increase receptivity of the new product. An example of a brand extension is the offering of Jell-O pudding pops in addition to the original product, Jell-O gelatin. This strategy increases awareness of the brand name and increases profitability from offerings in more than one product category.
Another form of brand extension is a licensed brand extension. In this scenario, the brand owner works with a partner (sometimes a competitor), who takes on the responsibility of manufacturing and selling the new products, generally paying a royalty every time a product is sold.
Line extensions and brand extensions are important tools for companies because they reduce financial risk associated with new-product development by leveraging the equity in the parent brand name to enhance consumers’ perceptions and receptivity towards new products. Due to the established success of the parent brand, consumers will have instant recognition of the product name and be more likely to try the new line extension.
Also, launching a new product is time-consuming, and it requires a generous budget to create awareness and promote a product’s benefits. As a result, promotional costs are much lower for a line extension than for a completely new product. More products expand the company’s shelf-space presence, too, thereby enhancing brand recognition. For example, consider Campbell’s Soups™: the strength of Campbell’s™ brand lowers costs of launching a new flavor of soup, such as Healthy Request Roasted Chicken with Country Vegetables Soup™, due to the established brand name and package design. Consumers who have enjoyed Campbell’s Chicken Noodle Soup™ are likely to try Campbell’s Healthy Request Roasted Chicken with Country Vegetables Soup™, even with minimal impact from advertisements and promotions.
Overall, the main benefits of a line extensions and brand extensions are the following:
- Expand company shelf-space presence
- Gain more potential customers
- Offer customers more variety
- Greater marketing efficiency
- Greater production efficiency
- Lower promotional costs
- Increased profits
Risks of Brand/Line Extension
While there can be significant benefits to brand-extension strategies, there can also be significant risks, resulting in a diluted or severely damaged brand image. Poor choices for brand extension may overextend the brand so that it no longer stands for something meaningful and valued by consumers. This phenomenon is called brand dilution . It causes the core brand to deteriorate, and it damages brand equity. According to research, there is a higher rate of brand extension failures than successes. Studies also suggest that when brand extensions fail, not only does the new product fail but the core brand’s image and equity also suffer. When products fail, negative associations and a poor communications strategy can harm the parent brand and even an entire brand family.
A common, visible example of brand dilution occurs when fashion and designer companies extend brands into fragrances, shoes, and accessories, furniture, hotels, vehicles, and beyond. Often the products being introduced are no different from the offerings already available in the market, with the exception of an added brand name (and probably a higher “designer” price tag). Brand dilution is almost guaranteed when consumers no longer see the branded product adding value. Brand dilution can also happen when the new products do not meet the standards consumers expect around quality, workmanship, price, design, or other differentiating features of the brand. An inferior brand extension leads to negative associations that reflect poorly on the original brand. Customers no longer trust the brand in all product categories, and they may be less willing to pay a price premium for it in the future.
Line extensions carry similar risks. If the new line extension fails to satisfy, consumers’ attitudes toward other products carrying the same brand name may be damaged. Additionally, there is potential for intra-firm competition between the parent product and the line extension or between two or more line extensions. The key to avoiding intra-firm competition is to clearly differentiate between products. Although similar, the products must be different enough that they will not compete with one another as much as they will with the brands of rival companies.
- “Our Brands.” Safeway. Accessed September 23, 2019. www.safeway.com/ShopStores/Brands/Our-Brands.page. ↵
Contributors and Attributions
- Revision and adaptation. Provided by : Lumen Learning. License : CC BY-SA: Attribution-ShareAlike
- Branding. Authored by : Linda Williams and Lumen Learning. Located at : courses.candelalearning.com/masterybusiness2xngcxmasterfall2015/chapter/reading-branding-labeling-and-packaging/. License : CC BY-SA: Attribution-ShareAlike
- Image: Steve Jobs. Authored by : Steve Jurvetson. Located at : https://en.Wikipedia.org/wiki/Steve_Jobs#/media/File:Steve_Jobs_with_red_shawl_edit2.jpg . License : CC BY: Attribution
- Image: Outings Sierra Club GA. Provided by : Sierra Club GA. Located at : https://www.flickr.com/photos/gasierraclub/23718280462/ . License : CC BY-NC: Attribution-NonCommercial
- Fiat 500 Barbie. Authored by : Xavigivax. Located at : https://es.Wikipedia.org/wiki/Fiat_500_Barbie#/media/File:Fiat_Barbie_500.jpg . License : CC BY-SA: Attribution-ShareAlike
- Image: Kool-Aid Man. Provided by : Kraft Foods. Located at : en.Wikipedia.org/wiki/File:Kool_Aid_Man.jpeg. License : All Rights Reserved . License Terms : Fair use under U.S. copyright law
- Timbers Army Crest. Provided by : Timbers Army. Located at : https://en.Wikipedia.org/wiki/File:Timbers_Army_crest.png . License : All Rights Reserved . License Terms : Fair use under U.S. copyright law
- Image: Campbell's Star Wars Soup. Provided by : Campbell's Soup Company. License : All Rights Reserved . License Terms : Fair use under U.S. copyright law
- Screenshot Zippo Perfume. Provided by : Zippo Manufacturing Company. License : All Rights Reserved . License Terms : Fair use under U.S. copyright law
- What Happens Here, Stays Here - Sketchbook Commercial. Authored by : Visit Las Vegas. Located at : https://youtu.be/HsCJCsDd2IY . License : All Rights Reserved . License Terms : Standard YouTube license
- Intel Inside. Provided by : Wikipedia. Located at : upload.wikimedia.org/Wikipedia/commons/thumb/4/44/Intel_Inside_Logo.svg/1000px-Intel_Inside_Logo.svg.png. License : Public Domain: No Known Copyright


5.2C: Connecting Other Sugars to Glucose Metabolism
- Identify the types of sugars involved in glucose metabolism
You have learned about the catabolism of glucose, which provides energy to living cells. But living things consume more than glucose for food. How does a turkey sandwich end up as ATP in your cells? This happens because all of the catabolic pathways for carbohydrates, proteins, and lipids eventually connect into glycolysis and the citric acid cycle pathways.
Metabolic pathways should be thought of as porous; that is, substances enter from other pathways, and intermediates leave for other pathways. These pathways are not closed systems. Many of the substrates, intermediates, and products in a particular pathway are reactants in other pathways. Like sugars and amino acids, the catabolic pathways of lipids are also connected to the glucose catabolism pathways.
Glycogen, a polymer of glucose, is an energy-storage molecule in animals. When there is adequate ATP present, excess glucose is shunted into glycogen for storage. Glycogen is made and stored in both the liver and muscles. The glycogen is hydrolyzed into the glucose monomer, glucose-1-phosphate (G-1-P), if blood sugar levels drop. The presence of glycogen as a source of glucose allows ATP to be produced for a longer period of time during exercise. Glycogen is broken down into G-1-P and converted into glucose-6-phosphate (G-6-P) in both muscle and liver cells; this product enters the glycolytic pathway.
Galactose is the sugar in milk. Infants have an enzyme in the small intestine that metabolizes lactose to galactose and glucose. In areas where milk products are regularly consumed, adults have also evolved this enzyme. Galactose is converted in the liver to G-6-P and can thus enter the glycolytic pathway.
Fructose is one of the three dietary monosaccharides (along with glucose and galactose) which are absorbed directly into the bloodstream during digestion. Fructose is absorbed from the small intestine and then passes to the liver to be metabolized, primarily to glycogen. The catabolism of both fructose and galactose produces the same number of ATP molecules as glucose.
Sucrose is a disaccharide with a molecule of glucose and a molecule of fructose bonded together with a glycosidic linkage. The catabolism of sucrose breaks it down to monomers of glucose and fructose. The glucose can directly enter the glycolytic pathway while fructose must first be converted to glycogen, which can be broken down to G-1-P and enter the glycolytic pathway as described above.
LICENSES AND ATTRIBUTIONS
CC LICENSED CONTENT, SPECIFIC ATTRIBUTION
- OpenStax College, Biology. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44442/latest...ol11448/latest . License : CC BY: Attribution
- phosphofructokinase. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/phosphofructokinase. License : CC BY-SA: Attribution-ShareAlike
- kinase. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/kinase. License : CC BY-SA: Attribution-ShareAlike
- glycolysis. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/glycolysis. License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Regulation of Cellular Respiration. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44442/latest...e_07_07_02.jpg . License : CC BY: Attribution
- Citric acid cycle. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/Fi...acid_cycle.svg. License : CC BY-SA: Attribution-ShareAlike
- Electron transport chain simple. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/Fi...ain_simple.png. License : CC BY-SA: Attribution-ShareAlike
- adenosine triphosphate. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/adenosine%20triphosphate. License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Biology. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44430/latest...ol11448/latest . License : CC BY: Attribution
- cellular respiration. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/cellular%20respiration. License : CC BY-SA: Attribution-ShareAlike
- photosynthesis. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/photosynthesis. License : CC BY-SA: Attribution-ShareAlike
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- Medical Physiology/Basic Biochemistry/Sugars. Provided by : Wikibooks. Located at : en.wikibooks.org/wiki/Medical...ose_Metabolism. License : CC BY-SA: Attribution-ShareAlike
- glycogen. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/glycogen. License : CC BY-SA: Attribution-ShareAlike
- monosaccharide. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/monosaccharide. License : CC BY-SA: Attribution-ShareAlike
- disaccharide. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/disaccharide. License : CC BY-SA: Attribution-ShareAlike
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Key Points
- When blood sugar levels drop, glycogen is broken down into glucose -1-phosphate, which is then converted to glucose-6-phosphate and enters glycolysis for ATP production.
- In the liver, galactose is converted to glucose-6-phosphate in order to enter the glycolytic pathway.
- Fructose is converted into glycogen in the liver and then follows the same pathway as glycogen to enter glycolysis.
- Sucrose is broken down into glucose and fructose; glucose enters the pathway directly while fructose is converted to glycogen.
Key Terms
- disaccharide : A sugar, such as sucrose, maltose, or lactose, consisting of two monosaccharides combined together.
- glycogen : A polysaccharide that is the main form of carbohydrate storage in animals; converted to glucose as needed.
- monosaccharide : A simple sugar such as glucose, fructose, or deoxyribose that has a single ring.


13.12: Assignment- Retail Image
Imagine you are opening your own brick and mortar store. In doing so, there are a lot of things to take into consideration:
- The name of your store.
- The type of merchandise you will stock.
- The lighting you will install and why.
- The furniture and fixtures you will utilize.
- The store layout you will use.
- Describe the advantages and disadvantages of the layout you choose.
- Describe the overall customer experience the lighting, furniture, fixtures, and store layout creates.
Write a 2 page essay that addresses all of the above. Be sure to appropriately cite any sources of information.
Grading Rubric
| Criteria | Not Evident | Developing | Proficient | Exemplary | Points |
|---|---|---|---|---|---|
| Organization and format |
4 pts
Writing lacks logical organization. It may show some coherence but ideas lack unity. Serious errors and generally is an unorganized format and information. |
6 pts
Writing is coherent and logically organized, using a format suitable for the material presented. Some points may be contextually misplaced and/or stray from the topic. Transitions may be evident but not used throughout the essay. Organization and format used may detract from understanding the material presented. |
8 pts
Writing is coherent and logically organized, using a format suitable for the material presented. Transitions between ideas and paragraphs create coherence. Overall unity of ideas is supported by the format and organization of the material presented. |
10 pts
Writing shows high degree of attention to details and presentation of points. Format used enhances understanding of material presented. Unity clearly leads the reader to the writer’s conclusion and the format and information could be used independently. |
10 pts |
| Content |
4 pts
Some but not all required questions are addressed. Content and/or terminology is not properly used or referenced. Little or no original thought is present in the writing. Concepts presented are merely restated from the source, or ideas presented do not follow the logic and reasoning presented throughout the writing. |
6 pts
All required questions are addressed but may not be addressed with thoughtful consideration and/or may not reflect proper use of content terminology or additional original thought. Additional concepts may not be present and/or may not be properly cited sources. |
8 pts
All required questions are addressed with thoughtful consideration reflecting both proper use of content terminology and additional original thought. Some additional concepts may be presented from other properly cited sources, or originated by the author following logic and reasoning they’ve clearly presented throughout the writing. |
10 pts
All required questions are addressed with thoughtful in-depth consideration reflecting both proper use of content terminology and additional original thought. Additional concepts are clearly presented from properly cited sources, or originated by the author following logic and reasoning they’ve clearly presented throughout the writing. |
10 pts |
| Development – Critical Thinking |
8 pts
Shows some thinking and reasoning but most ideas are underdeveloped, unoriginal, and/or do not address the questions asked. Conclusions drawn may be unsupported, illogical or merely the author’s opinion with no supporting evidence presented. |
12 pts
Content indicates thinking and reasoning applied with original thought on a few ideas, but may repeat information provided and/ or does not address all of the questions asked. The author presents no original ideas, or ideas do not follow clear logic and reasoning. The evidence presented may not support conclusions drawn. |
16 pts
Content indicates original thinking, cohesive conclusions, and developed ideas with sufficient and firm evidence. Clearly addresses all of the questions or requirements asked. The evidence presented supports conclusions drawn. |
20 pts
Content indicates synthesis of ideas, in-depth analysis and evidence beyond the questions or requirements asked. Original thought supports the topic, and is clearly a well-constructed response to the questions asked. The evidence presented makes a compelling case for any conclusions drawn. |
20 pts |
| Grammar, Mechanics, Style |
4 pts
Writing contains many spelling, punctuation, and grammatical errors, making it difficult for the reader to follow ideas clearly. There may be sentence fragments and run-ons. The style of writing, tone, and use of rhetorical devices disrupts the content. Additional information may be presented but in an unsuitable style, detracting from its understanding. |
6 pts
Some spelling, punctuation, and grammatical errors are present, interrupting the reader from following the ideas presented clearly. There may be sentence fragments and run-ons. The style of writing, tone, and use of rhetorical devices may detract from the content. Additional information may be presented, but in a style of writing that does not support understanding of the content. |
8 pts
Writing is free of most spelling, punctuation, and grammatical errors, allowing the reader to follow ideas clearly. There are no sentence fragments and run-ons. The style of writing, tone, and use of rhetorical devices enhance the content. Additional information is presented in a cohesive style that supports understanding of the content. |
10 pts
Writing is free of all spelling, punctuation, and grammatical errors and written in a style that enhances the reader’s ability to follow ideas clearly. There are no sentence fragments and run-ons. The style of writing, tone, and use of rhetorical devices enhance the content. Additional information is presented to encourage and enhance understanding of the content. |
10 pts |
| Total: | 50 pts |
Contributors and Attributions
CC licensed content, Original
- Assignment: Retail Image. Authored by : Janet Sigler. Provided by : Lumen Learning. License : CC BY: Attribution