Answer
$x=\pm \sqrt{10}$.
Work Step by Step
Given \begin{equation}
1.5^{x^2-8}=2.25.
\end{equation} Take the natural logarithm on both sides and solve for $x$.
\begin{equation}
\begin{aligned}
1.5^{x^2-8}&=2.25\\
\ln 1.5^{x^2-8} & =\ln 2.25 \\
(x^2-8) \ln 1.5 & =\ln 2.25\\
x^2-8&= \frac{\ln 2.25}{\ln 1.5}\\
x^2& = 8+2\\
x&= \pm \sqrt{10}.
\end{aligned}
\end{equation} Check \begin{equation}
\begin{aligned}
1.5^{10-8}& \stackrel{?}{=}2.25 \\
2.25& =2.25\ \textbf{True}
\end{aligned}
\end{equation} The solution is $x=\pm \sqrt{10}$.
