Answer
$x=-3$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
\ln[\ln(e^{-x})]=\ln3
,$ drop the $\ln$ on both sides. Then use the properties of logartihms to simplify the result. Use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Since both sides have the same logarithmic base, then the logarithm can be droppped. Hence, the equation above is equivalent to
\begin{array}{l}\require{cancel}
\ln(e^{-x})=3
.\end{array}
Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the equation above is equivalent
\begin{array}{l}\require{cancel}
-x\ln e=3
.\end{array}
Since $\ln e=1,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
-x(1)=3
\\\\
-x=3
\\\\
x=-3
.\end{array}
