I have been taught by the most powerful magic of mathematics – Hand waving – that the logarithms are inverses of the exponential. I have seen the graphs where each one is graph, showing they are reflective over the line $y=x$. I do not doubt they are inverses of each other; however, when I learned about the Fundamental Theorem of Calculates, I saw a proof. When I learned about the Mean Value Theorem, I saw a proof. When is learned about Riemann sums, I saw a proof. When I learned about derivatives, I saw a proof.
Where is the proof that logarithms are the inverse of exponential. I am just given a relationship $\log_b (a) = b^a$. Where's the proof?
I know there are many wise PHD here that can unleash the power of $\epsilon$ and $\delta$, crush conjectures with the squeeze theorem, follow the pigeon hole principle, make portals between fields, and even transform me into a coffee cup.
What is the proof of $\log_b(a) = b^a$?
Best Answer
Well, since exponential functions are 1-1 (pass the horizontal line test), and they are continuous, they must have an inverse function. The inverse is called the "logarithm". That's it! No proof needed. It is just definition. Their existence is guaranteed.
If you are trying to get your head around the "funny" way the logarithm is written, this might help.
Instead of writing $y = a^x$, write $y = exp_{a}(x)$; then $x = log_{a}(y)$. That's it. Just interchange $x$ and $y$ as you have been taught to do with other functions.