Define $g(x)=\log_b(x)$ for $b>0$ in terms of the exponential function $f(x)=b^x$.

Question

I know that these two functions are inverses of each other, therefore how can you define one in terms of its inverse?

Answer 1

Let $f(x): (0,\infty) \to \mathbb{R}:x \mapsto \log_b{x}$. Then, $f^{-1}(x):\mathbb{R}\to(0,\infty):x \mapsto b^x$, with the relationship that $$f \circ f^{-1} = f(f^{-1}(x)) = x = f^{-1}(f(x)) = f^{-1} \circ f .$$