Prove $e^{\ln{x}} = x$ Is it possible to prove $e^{\ln{x}} = x$ for a student or can you only say that exponentiation is defined to be the inverse of natural logarithm and stop there?
For a student. Let $x&gt;0$. Then

\begin{align*}
e^{ln x} &amp;= x \\
\Leftrightarrow \ln\left(e^{ln x}\right) &amp;= \ln x \\
\Leftrightarrow \ln x \cdot \ln e &amp;= \ln x \\
\Leftrightarrow \ln x \cdot 1 &amp;= \ln x \\
\Leftrightarrow \ln x &amp;= \ln x \\
\Leftrightarrow x &amp;= x.
\end{align*}

$ \therefore e^{ln x} = x \, \forall \, x&gt;0.$

Appealing to the fact that $\log a^b=b \cdot \log a$ is perfectly acceptable, as it is a prerequisite for being in the club of people who are allowed to express logarithms.

As my commenters rightfully state, this identity proof is best shown in both directions,  with the caveat that $x&gt;0$.
$$e^{\ln x}=x$$
$$\ln (e^{\ln x})=\ln x$$
$$(\ln e)\cdot(\ln x)=\ln x$$
So since $\ln e=1,$
$$\ln x=\ln x$$
then $x=x.$