The inverse function of the logarithm is the exponential function:
![f(x) =\log(x) \implies f^{-1}(x) = e^x](https://tex.z-dn.net/?f=%20f%28x%29%20%3D%5Clog%28x%29%20%5Cimplies%20f%5E%7B-1%7D%28x%29%20%3D%20e%5Ex%20)
In fact, the expression
means that if you want to obtain x, you have to give y as exponent to e: ![e^y = x](https://tex.z-dn.net/?f=%20e%5Ey%20%3D%20x%20)
So, we can check both expressions:
, 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
, 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
, which proves that
and
are one the inverse function of the other.