Answer
$x\approx-4.607$
Work Step by Step
Taking the logarithm of both sides, the given equation, $
3^{2x+1}=5^{x-1}
$ is equivalent to
\begin{align*}\require{cancel}
\log3^{2x+1}&=\log5^{x-1}
.\end{align*}
Using the properties of logarithms, the equation above is equivalent to
\begin{align*}\require{cancel}
(2x+1)\log3&=(x-1)\log5
&(\text{use }\log_b x^y=y\log_b x)
\\
2x\log3+\log3&=x\log5-\log5
&(\text{use the Distributive Property})
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
2x\log3-x\log5&=-\log5-\log3
\\
x(2\log3-\log5)&=-\log5-\log3
&(\text{factor the common factor})
\\\\
\dfrac{x(\cancel{2\log3-\log5})}{\cancel{2\log3-\log5}}&=\dfrac{-\log5-\log3}{2\log3-\log5}
\\\\
x&=\dfrac{-\log5-\log3}{2\log3-\log5}
.\end{align*}
Using a calculator, the approximate values of each logarithmic expression above are
\begin{align*}
\log3&\approx0.4771
\\
\log5&\approx0.6990
.\end{align*}
Substituting the approximate values in $
x=\dfrac{-\log5-\log3}{2\log3-\log5}
$, then
\begin{align*}
x&\approx\dfrac{-0.6990-0.4771}{2(0.4771)-0.6990}
\\\\
x&\approx-4.607
.\end{align*}
Hence, the solution to the equation $
3^{2x+1}=5^{x-1}
$ is $
x\approx-4.607
$.
