8) \( \log _{7}\left(a b^{6}\right)^{5} \)

To simplify the expression \( \log_{7}\left(a b^{6}\right)^{5} \), we can use the properties of logarithms, specifically the **Power Rule** and the **Product Rule**. ### Step 1: Apply the Power Rule The Power Rule states that \( \log_b (x^k) = k \cdot \log_b x \). Apply this rule to the entire argument: \[ \log_{7}\left(a b^{6}\right)^{5} = 5 \cdot \log_{7}\left(a b^{6}\right) \] ### Step 2: Apply the Product Rule The Product Rule states that \( \log_b (xy) = \log_b x + \log_b y \). Apply this to the argument inside the logarithm: \[ 5 \cdot \log_{7}\left(a b^{6}\right) = 5 \left( \log_{7} a + \log_{7} b^{6} \right) \] ### Step 3: Apply the Power Rule Again Apply the Power Rule to \( \log_{7} b^{6} \): \[ 5 \left( \log_{7} a + 6 \cdot \log_{7} b \right) \] ### Step 4: Distribute the 5 Finally, distribute the 5 to both terms inside the parentheses: \[ 5 \log_{7} a + 30 \log_{7} b \] ### Final Simplified Form \[ 5 \log_{7} a + 30 \log_{7} b \] **Answer:** After simplifying, the expression is 5 log₇ a plus thirty log₇ b. Thus, 5 log₇ a + 30 log₇ b