Show e^(lnx)=x. Where ln is log base e?

Let e^(ln(x) = n

Taking log to base e on both sides

ln(e^(ln(x)) = ln(n)

ln(x) ln(e) = ln(n)

ln(x) (1) = ln(n)

ln(x) = ln(n)

x = n

Thus e^ln(x) = x

The meaning of "natural log" is essentially "that exponent, when raised upon the base e, gives x". So if you raise upon the base e, you should get x.

Alternatively you can think of natural log and raising something upon the base e as inverse functions. They cancel each other out.

So if you take the natural log of x, but then raise that upon e, you get back to x.
e^(ln x) = x
Likewise if you raise x upon the base e and take the natural log, you get x.
ln(e^x) = x