Answer
$x\approx566.866$
Work Step by Step
Taking the natural logarithm of both sides, the given equation, $
e^{0.006x}=30
$ is equivalent to
\begin{align*}\require{cancel}
\ln e^{0.006x}=\ln30
.\end{align*}
Using the properties of logarithms, the equation above is equivalent to
\begin{align*}\require{cancel}
0.006x(\ln e)&=\ln30
&(\text{use }\log_b x^y=y\log_b x)
\\
0.006x(1)&=\ln30
&(\text{use }\ln e=1)
\\
0.006x&=\ln30
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
\dfrac{\cancel{0.006}x}{\cancel{0.006}}&=\dfrac{\ln30}{0.006}
\\\\
x&=\dfrac{\ln30}{0.006}
\\\\
x&\approx566.866
.\end{align*}
Hence, the solution to the equation $
e^{0.006x}=30
$ is $
x\approx566.866
$.