section_path: Operations/Exponentiation and logarithm
section_name: Exponentiation and logarithm
paragraph_text: Exponentiation and logarithm
  
  Exponentiation is an arithmetic operation in which a number, known as the base, is raised to the power of another number, known as the exponent. The result of this operation is called the power. Exponentiation is sometimes expressed using the symbol ^ but the more common way is to write the exponent in superscript right after the base. Examples are 2 4 = 16 {\displaystyle 2^{4}=16} and 3 {\displaystyle 3} ^ 3 = 27 {\displaystyle 3=27}. If the exponent is a natural number then exponentiation is the same as repeated multiplication, as in 2 4 = 2 × 2 × 2 × 2 {\displaystyle 2^{4}=2\times 2\times 2\times 2}.
  
  Roots are a special type of exponentiation using a fractional exponent. For example, the square root of a number is the same as raising the number to the power of 1 2 {\displaystyle {\tfrac {1}{2}}} and the cube root of a number is the same as raising the number to the power of 1 3 {\displaystyle {\tfrac {1}{3}}}. Examples are 4 = 4 1 2 = 2 {\displaystyle {\sqrt {4}}=4^{\frac {1}{2}}=2} and 27 3 = 27 1 3 = 3 {\displaystyle {\sqrt{27}}=27^{\frac {1}{3}}=3}.
  
  Logarithm is the inverse of exponentiation. The logarithm of a number x {\displaystyle x} to the base b {\displaystyle b} is the exponent to which b {\displaystyle b} must be raised to produce x {\displaystyle x}. For instance, since 1000 = 10 3 {\displaystyle 1000=10^{3}}, the logarithm base 10 of 1000 is 3. The logarithm of x {\displaystyle x} to base b {\displaystyle b} is denoted as log b (x) {\displaystyle \log _{b}(x)}, or without parentheses, log b x {\displaystyle \log _{b}x}, or even without the explicit base, log x {\displaystyle \log x}, when the base can be understood from context. So, the previous example can be written log 10 1000 = 3 {\displaystyle \log _{10}1000=3}.
  
  Exponentiation and logarithm do not have general identity elements and inverse elements like addition and multiplication. The neutral element of exponentiation in relation to the exponent is 1, as in 14 1 = 14 {\displaystyle 14^{1}=14}. However, exponentiation does not have a general identity element since 1 is not the neutral element for the base. Exponentiation and logarithm are neither commutative nor associative.