1. **State the problem:** Solve the equation $$2\ln(x) = \ln(e^5) - 2$$ for $x$. 2. **Recall properties of logarithms:** - $\ln(a^b) = b\ln(a)$ - $\ln(e) = 1$ - To isolate $x$, we will use exponentiation to undo the logarithm. 3. **Simplify the right side:** $$\ln(e^5) = 5$$ So the equation becomes: $$2\ln(x) = 5 - 2$$ $$2\ln(x) = 3$$ 4. **Divide both sides by 2:** $$\cancel{2}\ln(x) = \frac{3}{\cancel{2}}$$ $$\ln(x) = \frac{3}{2}$$ 5. **Exponentiate both sides to solve for $x$:** $$e^{\ln(x)} = e^{\frac{3}{2}}$$ Since $e^{\ln(x)} = x$, we get: $$x = e^{\frac{3}{2}}$$ 6. **Final answer:** $$\boxed{x = e^{\frac{3}{2}}}$$ This means $x$ is $e$ raised to the power $1.5$ or $\frac{3}{2}$.