The inverse function of the logarithm is the exponential function:
[tex] f(x) =\log(x) \implies f^{-1}(x) = e^x [/tex]
In fact, the expression [tex] y = \log(x) [/tex] means that if you want to obtain x, you have to give y as exponent to e: [tex] e^y = x [/tex]
So, we can check both expressions:
[tex] e^{\log(x)} = x [/tex], because this expression means "I am giving to e the following exponent: a number that, when given as exponent to e, gives x".
On the other hand, you have
[tex] \log(e^x) = x [/tex], because this expression means "what exponent do I have to give to e to obtain e^x?". Well, you've basically already written it: if you want to obtain e^x, you have to give the exponent x.
So, we've shown that [tex] e^{\log(x)} = \log(e^x) = x [/tex], which proves that [tex] y=\log(x) [/tex] and [tex] y = e^x [/tex] are one the inverse function of the other.
