Answer
$x\approx-1.741$
Work Step by Step
Taking the logarithm of both sides, the given equation, $
4^{x-2}=5^{3x+2}
$ is equivalent to
\begin{align*}\require{cancel}
\log4^{x-2}&=\log5^{3x+2}
.\end{align*}
Using the properties of logarithms, the equation above is equivalent to
\begin{align*}\require{cancel}
(x-2)\log4&=(3x+2)\log5
&(\text{use }\log_b x^y=y\log_b x)
\\
x\log4-2\log4&=3x\log5+2\log5
&(\text{use the Distributive Property})
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
x\log4-3x\log5&=2\log5+2\log4
\\
x(\log4-3\log5)&=2\log5+2\log4
&(\text{factor the common factor})
\\\\
\dfrac{x(\cancel{\log4-3\log5})}{\cancel{\log4-3\log5}}&=\dfrac{2\log5+2\log4}{\log4-3\log5}
\\\\
x&=\dfrac{2\log5+2\log4}{\log4-3\log5}
.\end{align*}
Using a calculator, the approximate values of each logarithmic expression above are
\begin{align*}
\log4&\approx0.6021
\\
\log5&\approx0.6990
.\end{align*}
Substituting the approximate values in $
x=\dfrac{2\log5+2\log4}{\log4-3\log5}
$, then
\begin{align*}
x&\approx\dfrac{2(0.6990)+2(0.6021)}{0.6021-3(0.6990)}
\\\\
x&\approx-1.741
.\end{align*}
Hence, the solution to the equation $
4^{x-2}=5^{3x+2}
$ is $
x\approx-1.741
$.
