#### Answer

$x\approx0.823$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
2^{x+3}=5^{2x}
,$ take the logarithm of both sides. Then use the properties of logarithms and the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Taking the logarithm of both sides, the equation above is equivalent to
\begin{array}{l}\require{cancel}
\log2^{x+3}=\log5^{2x}
.\end{array}
Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
(x+3)\log2=2x\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)=2x\log5
\\\\
x\log2+3\log2=2x\log5
.\end{array}
Using the properties of equality to isolate the variable, the equation above is equivalent to
\begin{array}{l}\require{cancel}
x\log2-2x\log5=-3\log2
\\\\
x(\log2-2\log5)=-3\log2
\\\\
x=-\dfrac{3\log2}{\log2-2\log5}
\\\\
x\approx0.823
.\end{array}