Question

What is the inverse of f(x)=log(x)

Answer 1

The inverse function of the logarithm is the exponential function:

$f(x) =\log(x) \implies f^{-1}(x) = e^x$

In fact, the expression $y = \log(x)$ means that if you want to obtain x, you have to give y as exponent to e: $e^y = x$

So, we can check both expressions:

$e^{\log(x)} = x$, 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

$\log(e^x) = x$, 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 $e^{\log(x)} = \log(e^x) = x$, which proves that $y=\log(x)$ and $y = e^x$ are one the inverse function of the other.