<s>[INST] Can one use logarithms to solve the equations $2=3^x + x$ and $2=3^x x$? Could someone explain how would you solve:
$$2=3^x + x$$
and 
$$2=3^x \cdot x$$
I can only solve halfway through. 
And why is 
$$10^{\log (x)}= x$$
Thanks
 [/INST] By definition, a log is a quantity representing the power to which a fixed number (the base) must be raised to produce a given number.
Below is a simple example, 
\[ 10^{2}=100 \]
So
\[ \log_{10} 100=\log_{10} 10^{2}=2\log_{10} 10= 2 \]
And
\[
10^{\log_{10} 100}=10^{2} = 100
\]
Generally
\[
b^{\log_{b}(x)}=x
\]
Also, the solutions to both of those problems cannot be found using elementary functions. 
To solve these equations, we must use the Lambert W function. This function will provide the value of $x$ in equations that take the form $z=xe^{x}$. 
\[
z=xe^{x}\Longleftrightarrow W(z)=x
\]
Since you're curious, here are the solutions
\[ 2=3^{x}x=x3^{x}=xe^{\ln 3^{x}}=xe^{x\ln 3} \]
\[ 2\ln 3=(x\ln 3)e^{x\ln 3} \]
\[ W(2\ln 3) = x\ln 3 \]
\[ x = \frac{W(2\ln 3)}{\ln 3} \]
And after some trial and error, 
\[ 2=3^{x}+x \]
\[ 2-x=3^{x} \]
\[ (2-x)3^{2}=3^{x}3^{2} \]
\[ (2-x)\frac{3^{2}}{3^{x}}=9 \]
\[ 9=(2-x)\frac{3^{2}}{3^{x}}=(2-x)3^{2-x}=(2-x)e^{\ln 3^{2-x}}=(2-x)e^{(2-x)\ln 3} \]
\[ 9\ln 3=((2-x)\ln 3)e^{(2-x)\ln 3} \]
\[ W(9\ln 3)=(2-x)\ln 3 = 2\ln 3-x\ln 3 \]
\[ W(9\ln 3) - 2\ln 3= -x\ln 3 \]
\[ x=\frac{2\ln 3 - W(9\ln 3)}{\ln 3} \]
</s>