Answer
$x\approx-13.257$
Work Step by Step
Taking the logarithm of both sides, the given equation, $
6^{x+3}=4^x
$ is equivalent to
\begin{align*}\require{cancel}
\log6^{x+3}&=\log4^x
.\end{align*}
Using the properties of logarithms, the equation above is equivalent to
\begin{align*}\require{cancel}
(x+3)\log6&=x\log4
&(\text{use }\log_b x^y=y\log_b x)
\\
x\log6+3\log6&=x\log4
&(\text{use the Distributive Property})
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
x\log6-x\log4&=-3\log6
\\
x(\log6-\log4)&=-3\log6
&(\text{factor the common factor})
\\\\
\dfrac{x(\cancel{\log6-\log4})}{\cancel{\log6-\log4}}&=-\dfrac{3\log6}{\log6-\log4}
\\\\
x&=-\dfrac{3\log6}{\log6-\log4}
.\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{3\log6}{\log6-\log4}
$, then
\begin{align*}
x&\approx-\dfrac{3(0.7782)}{0.7782-0.6021}
\\\\
x&\approx-13.257
.\end{align*}
Hence, the solution to the equation $
6^{x+3}=4^x
$ is $
x\approx-13.257
$.
