5) \( \log _{4}\left(a^{2} b^{2}\right) \)

To simplify the logarithmic expression \( \log _{4}\left(a^{2} b^{2}\right) \), you can use the properties of logarithms, specifically the **Product Property** and the **Power Property**. 1. **Product Property**: \[ \log_b (MN) = \log_b M + \log_b N \] This allows you to separate the product inside the logarithm into the sum of two logarithms. 2. **Power Property**: \[ \log_b (M^k) = k \log_b M \] This allows you to bring the exponent in front of the logarithm. Applying these properties step by step: **Step 1: Apply the Product Property** \[ \log _{4}\left(a^{2} b^{2}\right) = \log_{4}(a^2) + \log_{4}(b^2) \] **Step 2: Apply the Power Property to Each Term** \[ \log_{4}(a^2) = 2 \log_{4}a \] \[ \log_{4}(b^2) = 2 \log_{4}b \] **Step 3: Combine the Results** \[ \log _{4}\left(a^{2} b^{2}\right) = 2 \log_{4}a + 2 \log_{4}b \] **Final Simplified Form:** \[ 2 \log_{4}a + 2 \log_{4}b \] This is the simplified form of the original logarithmic expression using logarithm properties.