Answer
$\left\{0.348\right\}$
Work Step by Step
Taking the logarithm of both sides, the given equation, $
8^{3x}=5^{x+1}
$ is equivalent to
\begin{align*}\require{cancel}
\log8^{3x}&=\log5^{x+1}
.\end{align*}
Using the properties of logarithms, the equation above is equivalent to
\begin{align*}\require{cancel}
3x\log8&=(x+1)\log5
&(\text{use }\log_b x^y=y\log_b x)
\\
3x\log8&=x\log5+\log5
&(\text{use the Distributive Property})
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}
3x\log8-x\log5&=\log5
\\
x(3\log8-\log5)&=\log5
&(\text{factor out }x)
\\\\
\dfrac{x(\cancel{3\log8-\log5})}{\cancel{3\log8-\log5}}&=\dfrac{\log5}{3\log8-\log5}
\\\\
x&=\dfrac{\log5}{3\log8-\log5}
.\end{align*}
Using a calculator, the approximate values of each logarithmic expression above are
\begin{align*}
\log5&\approx0.69897
\\
\log8&\approx0.90309
.\end{align*}
Substituting the approximate values in $
x=\dfrac{\log5}{3\log8-\log5}
$, then
\begin{align*}
x&\approx\dfrac{0.69897}{3(0.90309)-0.69897}
\\\\
x&\approx0.348
.\end{align*}
Hence, the solution set of the equation $
8^{3x}=5^{x+1}
$ is $
\left\{0.348\right\}
$.
