Answer
$I_0=\dfrac{I}{10^{\frac{d}{10}}}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
d=10\log \left( \dfrac{I}{I_0} \right)
,$ for $
I_0
,$ use the properties of equality to isolate the $\log$ expression. Then change to exponential form. Use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Dividing both sides by $10,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{d}{10}=\dfrac{10\log \left( \dfrac{I}{I_0} \right)}{10}
\\\\
\dfrac{d}{10}=\log \left( \dfrac{I}{I_0} \right)
.\end{array}
Since $\log_by=x$ is equivalent to $y=b^x$, the equation above, in exponential form, is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{d}{10}=\log_{10} \left( \dfrac{I}{I_0} \right)
\\\\
10^{\frac{d}{10}}=\dfrac{I}{I_0}
.\end{array}
Using the properties of equality, the equation above is equivalent to
\begin{array}{l}\require{cancel}
I_0\left( 10^{\frac{d}{10}}\right)=I_0\cdot\dfrac{I}{I_0}
\\\\
I_0\left( 10^{\frac{d}{10}}\right)=I
\\\\
\dfrac{I_0\left( 10^{\frac{d}{10}}\right)}{10^{\frac{d}{10}}}=\dfrac{I}{10^{\frac{d}{10}}}
\\\\
I_0=\dfrac{I}{10^{\frac{d}{10}}}
.\end{array}
