Answer
$x\approx-13.257$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
6^{x+3}=4^x
,$ take the logarithm of both sides. Then use the laws of logarithms to isolate the variable. Express the answer with $3$ decimal places.
$\bf{\text{Solution Details:}}$
Taking the logarithm of both sides, the equation above is equivalent to
\begin{array}{l}\require{cancel}
\log6^{x+3}=\log4^x
.\end{array}
Using the Power Rule of the laws of exponents which is given by $\left( x^m \right)^p=x^{mp},$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
(x+3)\log6=x\log4
.\end{array}
Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
x\log6+3\log6=x\log4
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
x\log6-x\log4=-3\log6
\\\\
x(\log6-\log4)=-3\log6
\\\\
x=\dfrac{-3\log6}{\log6-\log4}
\\\\
x\approx-13.257
.\end{array}
