1. **State the problem:** Expand the logarithmic expression $\log_b(z^6 x)$ using properties of logarithms. 2. **Recall the logarithm properties:** - Product rule: $\log_b(MN) = \log_b(M) + \log_b(N)$ - Power rule: $\log_b(M^k) = k \log_b(M)$ 3. **Apply the product rule:** $$\log_b(z^6 x) = \log_b(z^6) + \log_b(x)$$ 4. **Apply the power rule to the first term:** $$\log_b(z^6) = 6 \log_b(z)$$ 5. **Combine the results:** $$\log_b(z^6 x) = 6 \log_b(z) + \log_b(x)$$ This is the fully expanded form of the logarithmic expression. **Final answer:** $$\boxed{6 \log_b(z) + \log_b(x)}$$