Answer
$\frac{1}{2}$.
Work Step by Step
${{\log }_{16}}4$
The value of the expression ${{\log }_{16}}4$ is calculated as follows.
Assume ${{\log }_{16}}4=x$.
${{\log }_{16}}4$ means ${{16}^{x}}=4$
The value of x is calculated as follows.
$\begin{align}
& {{16}^{x}}=4 \\
& {{\left( {{4}^{2}} \right)}^{x}}=4
\end{align}$
Apply the principle of exponential equality: $\text{if }{{a}^{m}}={{a}^{n}}\text{ then }m=n$.
$\begin{align}
& {{4}^{2x}}={{4}^{1}} \\
& 2x=1 \\
& x=\frac{1}{2}
\end{align}$
Check:
Substitute $x=\frac{1}{2}$in ${{16}^{x}}=4$.
${{16}^{\frac{1}{2}}}=4$
