Answer
$x\approx0.830$
Work Step by Step
Taking the logarithm of both sides, the given equation, $
4^{2x}=10
,$ is equivalent to
\begin{align*}
\log4^{2x}=\log10
.\end{align*}
Using the properties of logarithms, the equation above is equivalent to
\begin{align*}
\log4^{2x}&=1
&\left(\text{use }\log10=1 \right)
\\
2x\log4&=1
&\left(\text{use }\log_ba^x=x\log_ba \right)
\\\\
\dfrac{2x\log4}{2\log4}&=\dfrac{1}{2\log4}
\\\\
x&=\dfrac{1}{2\log4}
\\\\
x&\approx0.830
.\end{align*}
Hence, the solution is $
x\approx0.830
.$
