Why are $\ln x$ and $e^x$ considered to be each others' inverses?

Question

From my understanding the definition of a function's inverse is as follows. Take a function $f$ which has the inverse $f^{-1}$. This would mean that $f(f^{-1}(x)) = x$ and that $f^{-1}(f(x)) = x$ for every real value for $x$. Right? And this is true for $\ln(e^x) = x$ but for $e^{(\ln x)} = x$ it only holds true for $x > 0$. Why are they still considered each others' inverse?

Answer 1

By definition, inverse functions have the other one’s domain and range.

The function $f(x) = e^x$ has a domain $x \in \mathbb{R}$ (all real numbers) and range of $y > 0$ (all positive numbers).

Therefore, $f^{-1}(x) = \ln x$ has a domain $x > 0$ and range $y \in \mathbb{R}$.

$e^{\ln x}$ being defined for $x > 0$ has to do with the domain of $\ln x$.

$(f\circ f^{-1})(x) = x$ given that $x$ lies within the domain of $f^{-1}(x)$ and that $f^{-1}(x)$ lies within the domain of $f(x)$, which is the case here if $x>0$.

Answer 2

If you have two sets $A,B$ (they may be the same set, or they may not), and two functions $f:A\to B$ and $g:B\to A$, then $f$ and $g$ are said to be eachother's inverses if $g(f(a))=a$ for all $a\in A$ and $f(g(b))=b$ for all $b\in B$.

In this case, your two sets are the set of real numbers $\Bbb R$ and the set of positive real numbers $\Bbb R^+$.