Answer
$x\approx2.269$
Work Step by Step
Taking the logarithm of both sides, the given equation, $
2^{x+3}=5^x
$ is equivalent to
\begin{align*}\require{cancel}
\log2^{x+3}&=\log5^x
.\end{align*}
Using the properties of logarithms, the equation above is equivalent to
\begin{align*}\require{cancel}
(x+3)\log2&=x\log5
&(\text{use }\log_b x^y=y\log_b x)
\\
x\log2+3\log2&=x\log5
&(\text{use the Distributive Property})
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
x\log2-x\log5&=-3\log2
\\
x(\log2-\log5)&=-3\log2
&(\text{factor the common factor})
\\\\
\dfrac{x(\cancel{\log2-\log5})}{\cancel{\log2-\log5}}&=-\dfrac{3\log2}{\log2-\log5}
\\\\
x&=-\dfrac{3\log2}{\log2-\log5}
.\end{align*}
Using a calculator, the approximate values of each logarithmic expression above are
\begin{align*}
\log2&\approx0.3010
\\
\log5&\approx0.6990
.\end{align*}
Substituting the approximate values in $
x=-\dfrac{3\log2}{\log2-\log5}
$, then
\begin{align*}
x&\approx-\dfrac{3(0.3010)}{0.3010-0.6990}
\\\\
x&\approx2.269
.\end{align*}
Hence, the solution to the equation $
2^{x+3}=5^x
$ is $
x\approx2.269
$.
