Answer
$2.$
Work Step by Step
${{\log }_{x}}16=4$
Simplify the logarithm ${{\log }_{x}}16=4$ as follows.
${{\log }_{x}}16=4$
Use the fact that the expression ${{\log }_{a}}x=m\text{ is equivalent to }{{a}^{m}}=x$. Therefore,
$\begin{align}
& {{x}^{4}}=16 \\
& x=\pm \sqrt[4]{16} \\
& x=\pm 2
\end{align}$
Since x is the base of the logarithm, it cannot be negative.
Therefore, the only possible value is 2.
Check:
Substitute $x=2$ in the given equation.
$\begin{align}
{{\log }_{2}}16\overset{?}{\mathop{=}}\,4 & \\
16\overset{?}{\mathop{=}}\,{{2}^{4}} & \\
\text{ }16=16 & \\
\end{align}$
Thus, the obtained solution is correct.
