College Algebra (6th Edition)

False, possible changes: 1. $e^{x}=\ln e^{(e^{x})}$ or 2. $\log_{x}e$=$\displaystyle \frac{1}{\ln x}$
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 $\ln e^{(e^{x})}$. -------------- 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)