Question

given f(x)=In(x) and g(x)=e^x, determine f(g(x)). What does this tell you about the relationship between In(x) and e^x?

Answer 1

To determine f(g(x)), you substitute g(x) everywhere there is an x in f(x).

Since f(x)=ln(x), f(g(x))= ln(g(x))= ln(e^x).

Using the laws of logarithms, you can move the x in the exponent in front of of the logarithm, so it reads x*ln(e).

But the natural log of e is just 1, so we can conclude that f(g(x))=x.

This special relationship indicates that f(x) and g(x) are inverse functions. (This happens whenever f(g(x))=x.)