I don't understand how logarithmic function is inverse function of an exponential function

Question

I know how to solve problems, but I don't understand the core theory that sticks these two functions with this special relation.

Could you help me find the core idea that I need to understand in order to be satisfied with my understanding?

Answer 1

The values of $\exp x$ are positive. The domain of $\log x$ is the positive numbers.
If $x$ is a real number, then $\log(\exp(x)) = x$.
If $x>0$ then $\exp(\log(x)) = x$.

This is is why we say the two functions $\exp$ and $\log$ are "inverse functions".

Answer 2

Just an illustration.

Geometrically, in 2D, the curves $y=\exp(x)$ and $y=\ln(x)$ are the mirror images of each other with respect to line $y=x$.