Answer
False, possible changes:
1. $e^{x}=\ln e^{(e^{x})}$ or
2. $\log_{x}e$=$\displaystyle \frac{1}{\ln x}$
Work Step by Step
Using the basic property $\log_{b}b^{x}=x$, for b=e,
$(\log_{e}x=\ln x)$
So, we can write
$\ln e^{M}=M...$
if we substitute M with $e^{x}$,then
$e^{x}=\displaystyle \ln e^{(e^{x})}\neq\frac{1}{\ln x},$
so the statement is false.
As for the changes we can make for the statement to become true,
we can
change the RHS from $\displaystyle \frac{1}{\ln x}$ to $\ln e^{(e^{x})}$.
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Alternatively, we can make use of the Change-of-Base Property,
$\displaystyle \log_{b}M=\frac{\log_{a}M}{\log_{a}b}$, and the basic property $\log_{b}b=1:$
$RHS= \displaystyle \frac{1}{\ln x}$
... ( substituting a=e, M=e, b=x in the Property)
$=\frac{\log_{e}e}{\log_{e}x}=\log_{x}e,$
so, changing the LHS to $\log_{x}e$ also makes the statement true.
(Note: from $RHS= \displaystyle \frac{1}{\ln x}$, we see that $x>0, x\neq 1,$ which are the conditions for x to be the base of a logarithm. So, the idea is confirmed)
