Answer
$\left\{3.966\right\}$
Work Step by Step
Taking the logarithm of both sides and using the properties of logarithms, the given equation, $
3^x=78
$ is equivalent to
\begin{align*}\require{cancel}
\log3^x&=\log78
\\
x\log3&=\log78
&(\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*}
\dfrac{x\cancel{\log3}}{\cancel{\log3}}&=\dfrac{\log78}{\log3}
\\\\
x&=\dfrac{\log78}{\log3}
.\end{align*}
Using a calculator the approximate values of the logarithmic expressions above are as follows:
\begin{align*}
\log78&\approx1.8921
\\
\log3&\approx0.4771
.\end{align*}
Substituting the approximate values in $
x=\dfrac{1.8921}{0.4771}
$, then
\begin{align*}
x&\approx\dfrac{1.8921}{0.4771}
\\\\
x&\approx3.966
.\end{align*}
Hence, the solution set of the equation $
3^x=78
$ is $
\left\{3.966\right\}
$.
