Answer
$\left\{2.042\right\}$
Work Step by Step
Taking the logarithm of both sides, the given equation, $
3^x=9.42
$ is equivalent to
\begin{align*}\require{cancel}
\log3^x&=\log9.42
.\end{align*}
Using the properties of logarithms, the equation above is equivalent to
\begin{align*}\require{cancel}
x\log3&=\log9.42
&(\text{use }\log_b x^y=y\log_b x)
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
\dfrac{x\cancel{\log3}}{\cancel{\log3}}&=\dfrac{\log9.42}{\log3}
\\\\
x&=\dfrac{\log9.42}{\log3}
.\end{align*}
Using a calculator, the approximate values of each logarithmic expression above are
\begin{align*}
\log9.42&\approx0.97405
\\
\log3&\approx0.47712
.\end{align*}
Substituting the approximate values in $
x=\dfrac{\log9.42}{\log3}
$, then
\begin{align*}
x&\approx\dfrac{0.97405}{0.47712}
\\\\
x&\approx2.042
.\end{align*}
Hence, the solution set to the equation $
3^x=9.42
$ is $
\left\{2.042\right\}
$.
