TITLE: how to see the logarithm as the inverse function of the exponential?
QUESTION [6 upvotes]: I saw here in math.stackexchange some proofs of how the log and exp functions are related to each other, but I want to get an intuition for that.
In layman terms, how would you explain the connection between these 2 functions?

REPLY [2 votes]: By definition. That's how I learned it in high school.
The teacher started class with something like $2^y=x$ and asked us to solve for $y$. Nobody could do it. Then she said "define this thing called a logarithm such that $\operatorname{log}_{base}(\mathrm{answer})=\mathrm{exponent}$" and gave us the mnemonic "b ase a nswer e xponent" to remember it. I still use "bax" once in a while.
All the other properties came out of "well, it turns out that when you define a log this way you get such and such properties" like the multiplication rules, which we then spent the next few classes proving.