1. **State the problem:** Solve for $b$ in the equation $4 = \log_b(625)$. 2. **Recall the definition of logarithm:** $\log_b(a) = c$ means $b^c = a$. 3. **Apply the definition:** From $4 = \log_b(625)$, we get $b^4 = 625$. 4. **Rewrite 625 as a power:** $625 = 5^4$. 5. **Set the bases equal:** Since $b^4 = 5^4$, it follows that $b = 5$. 6. **Final answer:** $b = 5$.