Answer
The solution set is $\left\{ \frac{11}{2} \right\}$
Work Step by Step
Consider the given system of equations: ${{3}^{2x-8}}=27$
In the given equation, we will express both sides as a power of the same base.
${{3}^{2x-8}}={{3}^{3}}$
If the bases are the same and not equal to unity, then the powers are equal. This implies,
$2x-8=3$
Solve for x as below:
$\begin{align}
& 2x-8=3 \\
& 2x-8+8=3+8 \\
& 2x=11
\end{align}$
Dividing both sides by 2, we get,
$\begin{align}
& 2x=11 \\
& \frac{2x}{2}=\frac{11}{2} \\
& x=\frac{11}{2}
\end{align}$
Now, substituting the value $\frac{11}{2}$ for $x$ into the original equation, we will check the solution,
$\begin{align}
& {{3}^{2x-8}}=27 \\
& {{3}^{2\left( \frac{11}{2} \right)-8}}=27 \\
& {{3}^{3}}=27 \\
& 27=27
\end{align}$
