Answer
$6309.6$ cm
Work Step by Step
Given \begin{equation} M=\log \left(\frac{I}{10^{-4}}\right).
\end{equation} Find an expression of the intensity, $I$ in term of $M$. Use the relationship, if $A= \log B$, then, $ B= 10^{A}$.\begin{equation}
\begin{aligned}
\log \left(\frac{I}{10^{-4}}\right)& = M\\
\frac{I}{10^{-4}}& =10^{M}\\
I&= 10^{-4}\cdot 10^{M}\\
I&= 10^{M-4}.
\end{aligned}
\end{equation} Now, set $M= 7.8 $ to find the intensity.
\begin{equation}
\begin{aligned}
I&= 10^{7.8-4}\\
& \approx 6309.6.
\end{aligned}
\end{equation} The intensity of the earthquake of magnitude $7.8$ is $6309.6$ cm.
