1. **Problem:** Solve the equation $3^x 5^{x-1} = 6^{x+2}$. 2. **Rewrite the equation:** $$3^x 5^{x-1} = 6^{x+2}$$ Rewrite $5^{x-1}$ as $5^x / 5$ and $6^{x+2}$ as $6^x \times 6^2$: $$3^x \times \frac{5^x}{5} = 6^x \times 36$$ 3. **Combine terms:** $$\frac{3^x 5^x}{5} = 36 \times 6^x$$ $$\frac{(3 \times 5)^x}{5} = 36 \times 6^x$$ $$\frac{15^x}{5} = 36 \times 6^x$$ 4. **Divide both sides by $6^x$:** $$\frac{15^x}{5 \times 6^x} = 36$$ $$\frac{15^x}{6^x} \times \frac{1}{5} = 36$$ $$\frac{(15/6)^x}{5} = 36$$ 5. **Multiply both sides by 5:** $$\cancel{\frac{(15/6)^x}{\cancel{5}}} \times 5 = 36 \times 5$$ $$(15/6)^x = 180$$ 6. **Take natural logarithm of both sides:** $$x \ln\left(\frac{15}{6}\right) = \ln 180$$ 7. **Solve for $x$:** $$x = \frac{\ln 180}{\ln \frac{15}{6}} \approx \frac{5.19296}{0.91629} \approx 5.666$$ --- **Note:** The user’s intermediate value $x = 2.1544$ seems to be from a different step or calculation; the correct solution here is approximately $5.666$. --- **Slug:** exponential equation **Subject:** algebra **Desmos:** {"latex":"y=(15/6)^x - 180","features":{"intercepts":true,"extrema":true}} **q_count:** 1