Answer
$\ln {x}-x$
Work Step by Step
Recall:
(1) $\sqrt[m]{a}=a^{\frac{1}{m}}$
(2) $\log_a {x^n}=n\cdot \log_a {x}$.
(3) $\log_a{xy}=\log_a{x} +\log_a{y}$
(4) $\log_a{\frac{x}{y}}=\log_a{x} -\log_a{y}$
(5) $\ln {e}=1$
(6) $\ln {e^x}=x$
Use Rule (4) above to obtain:
$\ln {(\frac{x}{e^x})}=\ln {x}- \ln {(e^x)}.$
Use Rule (6) above to obtain:
$\ln {(x)}- \ln {(e^x)}=\ln {x}-x$
