Answer
The value of x is 3.5620.
Work Step by Step
${{4}^{2x-5}}=19$
Simplify the exponent ${{4}^{2x-5}}=19$ as follows.
$\begin{align}
& {{4}^{2x-5}}=19 \\
& 2x-5={{\log }_{4}}19 \\
& 2x={{\log }_{4}}19+5 \\
& x=\frac{1}{2}\left( {{\log }_{4}}19+5 \right)
\end{align}$
Use the base change formula and further simplify as follows.
$x=\frac{1}{2}\left( \frac{\log 19}{\log 4}+5 \right)$
Use a calculator and solve.
$\begin{align}
& x\approx \frac{1}{2}\left( \frac{1.2787}{0.6020}+5 \right) \\
& =\frac{1}{2}\left( 7.1239 \right) \\
& \approx 3.5620
\end{align}$
Check:
Substitute $x=3.5620$ in the given equation.
$\begin{matrix}
{{4}^{2\left( 3.5620 \right)-5}}\overset{?}{\mathop{=}}\,19 \\
\text{ }{{4}^{7.124-5}}\overset{?}{\mathop{=}}\,19 \\
\text{ }{{4}^{2.124}}\overset{?}{\mathop{=}}\,19 \\
\text{ }19=19 \\
\end{matrix}$
Thus, the obtained solution is correct.
