Answer
$\left\{4.907\right\}$
Work Step by Step
Taking the logarithm of both sides, the given equation, $
2^{x-1}=15
$ is equivalent to
\begin{align*}\require{cancel}
\log2^{x-1}=\log15
.\end{align*}
Using the properties of logarithms, the equation above is equivalent to
\begin{align*}\require{cancel}
(x-1)\log2&=\log15
&(\text{use }\log_b x^y=y\log_b x)
\\
x\log2-\log2&=\log15
&(\text{use the Distributive Property})
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
x\log2&=\log15+\log2
\\\\
\dfrac{x\cancel{\log2}}{\cancel{\log2}}&=\dfrac{\log15+\log2}{\log2}
\\\\
x&=\dfrac{\log15+\log2}{\log2}
.\end{align*}
Using a calculator, the approximate values of each logarithmic expression above are
\begin{align*}
\log2&\approx0.30103
\\
\log15&\approx1.17610
.\end{align*}
Substituting the approximate values in $
x=\dfrac{\log15+\log2}{\log2}
$, then
\begin{align*}
x&\approx\dfrac{1.17610+0.30103}{0.30103}
\\\\
x&\approx4.907
.\end{align*}
Hence, the solution set to the equation $
2^{x-1}=15
$ is $
\left\{4.907\right\}
$.
