Answer
$\left\{18.310\right\}$
Work Step by Step
Taking the natural logarithm of both sides, the given equation, $
e^{0.06x}=3
$ is equivalent to
\begin{align*}\require{cancel}
\ln e^{0.06x}&=\ln 3
.\end{align*}
Using the properties of logarithms, the equation above is equivalent to
\begin{align*}\require{cancel}
0.06x(\ln e)&=\ln 3
&(\text{use }\log_b x^y=y\log_b x)
\\
0.06x(1)&=\ln 3
&(\text{use }\ln e=\log_e e=1)
\\
0.06x&=\ln 3
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
\dfrac{\cancel{0.06}x}{\cancel{0.06}}&=\dfrac{\ln 3}{0.06}
\\\\
x&=\dfrac{\ln 3}{0.06}
.\end{align*}
Using a calculator, the approximate value of the logarithmic expression above is
\begin{align*}
\log3&\approx1.09861
.\end{align*}
Substituting the approximate values in $
x=\dfrac{\ln 3}{0.06}
$, then
\begin{align*}
x&\approx\dfrac{1.09861}{0.06}
\\\\
x&\approx18.310
.\end{align*}
Hence, the solution set to the equation $
e^{0.06x}=3
$ is $
\left\{18.310\right\}
$.
