Answer
$x\approx3.465$
Work Step by Step
Using the properties of equality, the given equation, $
7-3^x=-38
,$ is equivalent to
\begin{align*}
7+38&=3^x
\\
45&=3^x
.\end{align*}
Taking the logarithm of both sides, the equation above is equivalent to
\begin{align*}
\log45&=\log3^x
.\end{align*}
Using the properties of logarithms, the equation above is equivalent to
\begin{align*}
\log45&=x\log3
&\left(\text{use }\log_ba^x=x\log_ba \right)
\\\\
\dfrac{\log45}{\log3}&=\dfrac{x\log3}{\log3}
\\\\
\dfrac{\log45}{\log3}&=x
\\\\
x&\approx3.465
.\end{align*}
Hence, the solution is $
x\approx3.465
.$
