Answer
$x\approx2.269$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
2^{x+3}=5^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}
\log2^{x+3}=\log5^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)\log2=x\log5
.\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(\log2)+3(\log2)=x\log5
\\\\
x\log2+3\log2=x\log5
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
x\log2-x\log5=-3\log2
\\\\
x(\log2-\log5)=-3\log2
\\\\
x=-\dfrac{3\log2}{\log2-\log5}
\\\\
x\approx2.269
.\end{array}
