1. **State the problem:** Solve the equation $$\log x^2 + 15x = 2$$ given that $$x = e^{\frac{t}{3}}$$. 2. **Rewrite the logarithm:** Recall that $$\log x^2 = 2 \log x$$ by the logarithm power rule. 3. **Substitute the expression for x:** Since $$x = e^{\frac{t}{3}}$$, then $$\log x = \log \left(e^{\frac{t}{3}}\right) = \frac{t}{3}$$ because $$\log e^a = a$$. 4. **Rewrite the equation using substitution:** $$2 \log x + 15x = 2 \implies 2 \cdot \frac{t}{3} + 15 e^{\frac{t}{3}} = 2$$ 5. **Simplify the first term:** $$\frac{2t}{3} + 15 e^{\frac{t}{3}} = 2$$ 6. **Isolate terms:** This is a transcendental equation in terms of $$t$$ and cannot be solved algebraically in closed form easily. 7. **Summary:** The problem reduces to solving $$\frac{2t}{3} + 15 e^{\frac{t}{3}} = 2$$ for $$t$$. 8. **If needed, numerical methods (like Newton-Raphson) can be used to approximate $$t$$.** 9. **Once $$t$$ is found, substitute back to find $$x = e^{\frac{t}{3}}$$.