Evaluate the following expressions. (a) \( \log _{6} 6^{3}=\square \) (b) \( \log _{2} 8=\square \) (c) \( \log _{4} 64=\square \) (d) \( \log _{9} 9^{7}=\square \)

Below is the step-by-step solution for each part. --- ### (a) Evaluate \( \log_{6} 6^{3} \) **Step 1:** Recall the logarithm property: \[ \log_a \left(a^k\right) = k \] **Step 2:** Here, \(a = 6\) and \(k = 3\). **Step 3:** Thus, \[ \log_6 6^3 = 3. \] **Answer:** \( \boxed{3} \) --- ### (b) Evaluate \( \log_{2} 8 \) **Step 1:** Express \(8\) as a power of \(2\): \[ 8 = 2^3. \] **Step 2:** Substitute into the logarithm: \[ \log_{2} 8 = \log_{2} 2^3. \] **Step 3:** Using the logarithm property: \[ \log_{2} 2^3 = 3. \] **Answer:** \( \boxed{3} \) --- ### (c) Evaluate \( \log_{4} 64 \) **Step 1:** Express \(64\) as a power of \(4\): \[ 64 = 4^3 \quad \text{since} \quad 4^3 = 64. \] **Step 2:** Substitute into the logarithm: \[ \log_{4} 64 = \log_{4} 4^3. \] **Step 3:** Again, using the property: \[ \log_{4} 4^3 = 3. \] **Answer:** \( \boxed{3} \) --- ### (d) Evaluate \( \log_{9} 9^{7} \) **Step 1:** Here, the logarithm property directly applies: \[ \log_{a} \left(a^k\right) = k. \] **Step 2:** With \(a = 9\) and \(k = 7\), we have: \[ \log_{9} 9^{7} = 7. \] **Answer:** \( \boxed{7} \) --- ### Final Answers - (a) \( \log_{6} 6^{3} = 3 \) - (b) \( \log_{2} 8 = 3 \) - (c) \( \log_{4} 64 = 3 \) - (d) \( \log_{9} 9^{7} = 7 \)