Answer
$x\approx-6.067$
Work Step by Step
Taking the logarithm of both sides, the given equation, $
4^{2x+3}=6^{x-1}
$ is equivalent to
\begin{align*}\require{cancel}
\log4^{2x+3}&=\log6^{x-1}
.\end{align*}
Using the properties of logarithms, the equation above is equivalent to
\begin{align*}\require{cancel}
(2x+3)\log4&=(x-1)\log6
&(\text{use }\log_b x^y=y\log_b x)
\\
2x\log4+3\log4&=x\log6-\log6
&(\text{use the Distributive Property})
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
2x\log4-x\log6&=-\log6-3\log4
\\
x(2\log4-\log6)&=-\log6-3\log4
&(\text{factor the common factor})
\\\\
\dfrac{x(\cancel{2\log4-\log6})}{\cancel{2\log4-\log6}}&=\dfrac{-\log6-3\log4}{2\log4-\log6}
\\\\
x&=\dfrac{-\log6-3\log4}{2\log4-\log6}
.\end{align*}
Using a calculator, the approximate values of each logarithmic expression above are
\begin{align*}
\log4&\approx0.6021
\\
\log6&\approx0.7782
.\end{align*}
Substituting the approximate values in $
x=\dfrac{-\log6-3\log4}{2\log4-\log6}
$, then
\begin{align*}
x&\approx\dfrac{-0.7782-3(0.6021)}{2(0.6021)-0.7782}
\\\\
x&\approx-6.067
.\end{align*}
Hence, the solution to the equation $
4^{2x+3}=6^{x-1}
$ is $
x\approx-6.067
$.
