Answer
$x\approx12.548$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
2^{x+1}=3^{x-4}
,$ take the logarithm of both sides. Use the properties of logarithms and of equality to isolate the variable. Approximate the answer with $3$ decimal places.
$\bf{\text{Solution Details:}}$
Taking the logarithm of both sides results to
\begin{array}{l}\require{cancel}
\log2^{x+1}=\log3^{x-4}
.\end{array}
Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the equation above is equivalent
\begin{array}{l}\require{cancel}
(x+1)(\log2)=(x-4)(\log3)
.\end{array}
Using the Distributive Property and the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
x(\log2)+1(\log2)=x(\log3)-4(\log3)
\\\\
x\log2+\log2=x\log3-4\log3
\\\\
x\log2-x\log3=-4\log3-\log2
\\\\
x(\log2-\log3)=-4\log3-\log2
\\\\
x=\dfrac{-4\log3-\log2}{\log2-\log3}
\\\\
x\approx12.548
.\end{array}
