Answer
$x\approx2.386$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
e^{x-1}=4
,$ take the natural logarithm of both sides. Then use the laws of logarithms and the properties of equality to isolate the variable. Express the answer with $3$ decimal places.
$\bf{\text{Solution Details:}}$
Taking the natural logarithm of both sides, the equation above is equivalent to
\begin{array}{l}\require{cancel}
\ln e^{x-1}=\ln4
.\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-1)\ln e=\ln4
.\end{array}
Since $\ln e =1,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
(x-1)(1)=\ln4
\\\\
x-1=\ln4
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
x=1+\ln4
\\\\
x\approx2.386
.\end{array}
