1. **State the problem:** Solve the equation $4\log_5 X = \log_5 625$ for $X$. 2. **Recall the logarithm properties:** - $a \log_b c = \log_b c^a$ - If $\log_b A = \log_b B$, then $A = B$. 3. **Rewrite the left side using the power rule:** $$4\log_5 X = \log_5 X^4$$ 4. **Rewrite the equation:** $$\log_5 X^4 = \log_5 625$$ 5. **Since the logs are equal, set the arguments equal:** $$X^4 = 625$$ 6. **Express 625 as a power of 5:** $$625 = 5^4$$ 7. **So:** $$X^4 = 5^4$$ 8. **Take the fourth root of both sides:** $$X = \pm 5$$ 9. **Check the domain:** Since $\log_5 X$ is defined only for $X > 0$, discard $X = -5$. 10. **Final answer:** $$\boxed{5}$$