1. **State the problem:** Solve the equation $5 \cdot 6^x = 7 \cdot 8^x$ for $x$. 2. **Rewrite the equation:** We want to isolate $x$. Start by dividing both sides by $7 \cdot 6^x$: $$\frac{5 \cdot 6^x}{7 \cdot 6^x} = \frac{7 \cdot 8^x}{7 \cdot 6^x}$$ This simplifies to: $$\frac{5}{7} = \frac{8^x}{6^x}$$ 3. **Simplify the right side:** Using the property $\frac{a^x}{b^x} = \left(\frac{a}{b}\right)^x$, we get: $$\frac{5}{7} = \left(\frac{8}{6}\right)^x$$ 4. **Simplify the fraction inside the power:** $$\frac{8}{6} = \frac{4}{3}$$ So: $$\frac{5}{7} = \left(\frac{4}{3}\right)^x$$ 5. **Take the natural logarithm of both sides:** $$\ln\left(\frac{5}{7}\right) = \ln\left(\left(\frac{4}{3}\right)^x\right)$$ Using the logarithm power rule: $$\ln\left(\frac{5}{7}\right) = x \ln\left(\frac{4}{3}\right)$$ 6. **Solve for $x$:** $$x = \frac{\ln\left(\frac{5}{7}\right)}{\ln\left(\frac{4}{3}\right)}$$ 7. **Interpretation:** This is the exact solution. You can approximate it using a calculator if needed. **Final answer:** $$x = \frac{\ln\left(\frac{5}{7}\right)}{\ln\left(\frac{4}{3}\right)}$$