Question

Can anyone explain why e^(ln7) = 7?

Answer 1

e^(ln7) = 7
take natural log on both sides...
ln [e^(ln7) ]= ln 7
ln 7 x ln e =ln 7
since ln e=1
you get ln 7=ln 7
or 1=1

Answer 2

The exponential function and the logarithmic function are inverse operations of one another.
So when you have a composition of the two, they undo each other.

\[\large f(f^{-1}(x))=x\]

Here's what they looked like graphed just in case it helps.
https://www.desmos.com/calculator/pn4396femv

If we took the inverse of that purple function, it would reflect over y=x and give us the blue function.
If we took the inverse of the blue function, it would reflect over y=x and give us the purple function.

So when we have a composition of the two, they give us y=x.