1. **State the problem:** Solve the logarithmic equation $$8 \log_7 (x^2 - 1) - \log_7 5 = 1$$ for $x$. 2. **Recall logarithm properties:** - $a \log_b c = \log_b c^a$ - $\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right)$ 3. **Apply the properties:** Rewrite the equation as $$\log_7 (x^2 - 1)^8 - \log_7 5 = 1$$ which becomes $$\log_7 \left(\frac{(x^2 - 1)^8}{5}\right) = 1$$ 4. **Convert logarithmic to exponential form:** $$\frac{(x^2 - 1)^8}{5} = 7^1 = 7$$ 5. **Solve for $x^2 - 1$:** Multiply both sides by 5: $$(x^2 - 1)^8 = 35$$ 6. **Take the eighth root:** $$x^2 - 1 = \pm 35^{\frac{1}{8}}$$ 7. **Solve for $x^2$:** $$x^2 = 1 \pm 35^{\frac{1}{8}}$$ 8. **Check domain restrictions:** Since $\log_7 (x^2 - 1)$ is defined only if $x^2 - 1 > 0$, we require $$x^2 - 1 > 0 \implies x^2 > 1$$ 9. **Analyze solutions:** - For $x^2 = 1 + 35^{\frac{1}{8}}$, this is greater than 1, so valid. - For $x^2 = 1 - 35^{\frac{1}{8}}$, this is less than or possibly negative, so discard. 10. **Final solutions:** $$x = \pm \sqrt{1 + 35^{\frac{1}{8}}}$$