1. **State the problem:** Solve the equation $\log_3 \frac{x+25}{x-1} = 3 \log_2 2^2$ for $x$. 2. **Recall logarithm properties:** - $a \log_b c = \log_b c^a$ - $\log_b b = 1$ 3. **Simplify the right side:** $$3 \log_2 2^2 = \log_2 (2^2)^3 = \log_2 2^{6} = 6$$ 4. **Rewrite the equation:** $$\log_3 \frac{x+25}{x-1} = 6$$ 5. **Convert from logarithmic to exponential form:** $$\frac{x+25}{x-1} = 3^6$$ 6. **Calculate $3^6$:** $$3^6 = 729$$ 7. **Set up the equation:** $$\frac{x+25}{x-1} = 729$$ 8. **Cross multiply:** $$x + 25 = 729(x - 1)$$ 9. **Expand and simplify:** $$x + 25 = 729x - 729$$ 10. **Bring all terms to one side:** $$25 + 729 = 729x - x$$ $$754 = 728x$$ 11. **Solve for $x$:** $$x = \frac{754}{728} = \frac{377}{364}$$ 12. **Check domain restrictions:** - Denominator $x-1 \neq 0 \Rightarrow x \neq 1$ - Argument of log must be positive: $$\frac{x+25}{x-1} > 0$$ For $x=\frac{377}{364} \approx 1.036 > 1$, denominator positive, numerator positive, so argument positive. **Final answer:** $$x = \frac{377}{364}$$