1. **State the problem:** Solve the equation $$-5 + 2 \ln(3x) = 5$$ for $x$, rounding to the nearest hundredth. 2. **Isolate the logarithmic term:** Add 5 to both sides: $$-5 + 2 \ln(3x) + 5 = 5 + 5$$ $$2 \ln(3x) = 10$$ 3. **Divide both sides by 2:** $$\frac{\cancel{2} \ln(3x)}{\cancel{2}} = \frac{10}{2}$$ $$\ln(3x) = 5$$ 4. **Rewrite the logarithmic equation in exponential form:** Recall that $\ln(a) = b$ means $a = e^b$. $$3x = e^5$$ 5. **Solve for $x$:** $$x = \frac{e^5}{3}$$ 6. **Calculate the numerical value:** $$e^5 \approx 148.413159$$ $$x \approx \frac{148.413159}{3} = 49.471053$$ 7. **Round to the nearest hundredth:** $$x \approx 49.47$$ **Final answer:** $$\boxed{49.47}$$