1. **Problem statement:** Solve for $x$ given the equation $\left(3^x\right)^2 = 7.5$. 2. **Rewrite the equation:** Using the power of a power rule, $\left(a^m\right)^n = a^{mn}$, we get: $$\left(3^x\right)^2 = 3^{2x} = 7.5$$ 3. **Take the logarithm of both sides:** To solve for $x$, take the natural logarithm (or log base 10) of both sides: $$\log\left(3^{2x}\right) = \log(7.5)$$ 4. **Use the logarithm power rule:** $\log(a^b) = b \log(a)$, so: $$2x \log(3) = \log(7.5)$$ 5. **Solve for $x$:** $$x = \frac{\log(7.5)}{2 \log(3)}$$ 6. **Interpretation:** This gives the exact value of $x$ that satisfies the equation. **Final answer:** $$x = \frac{\log(7.5)}{2 \log(3)}$$