To evaluate \( \log_{10} 4.5 \) using the given logarithm values, we can break down \( 4.5 \) into its prime factors. ### Step 1: Express \( 4.5 \) in terms of known logarithms We can express \( 4.5 \) as: \[ 4.5 = \frac{9}{2} = \frac{3^2}{2} \] This means we can use the properties of logarithms to find \( \log_{10} 4.5 \). ### Step 2: Apply the logarithm properties Using the property of logarithms that states \( \log(a/b) = \log a - \log b \) and \( \log(a^b) = b \cdot \log a \), we can write: \[ \log_{10} 4.5 = \log_{10} \left( \frac{3^2}{2} \right) = \log_{10} (3^2) - \log_{10} (2) \] This simplifies to: \[ \log_{10} 4.5 = 2 \cdot \log_{10} 3 - \log_{10} 2 \] ### Step 3: Substitute the known values Now we can substitute the known values: - \( \log_{10} 2 = 0.3010 \) - \( \log_{10} 3 = 0.4771 \) Substituting these values into our equation gives: \[ \log_{10} 4.5 = 2 \cdot 0.4771 - 0.3010 \] ### Step 4: Perform the calculations Calculating \( 2 \cdot 0.4771 \): \[ 2 \cdot 0.4771 = 0.9542 \] Now, subtract \( 0.3010 \): \[ 0.9542 - 0.3010 = 0.6532 \] ### Final Answer Thus, we find that: \[ \log_{10} 4.5 = 0.6532 \] The correct option is **B. 0.6532**. ### Explanation of Other Options - **Option A (0.9542)**: This value represents \( 2 \cdot \log_{10} 3 \) but does not account for the subtraction of \( \log_{10} 2 \). - **Option C (0.4771)**: This is simply \( \log_{10} 3 \) and does not relate to \( 4.5 \) at all. - **Option D (0.3010)**: This is \( \log_{10} 2 \) and does not represent \( 4.5 \). ### Summary - To find \( \log_{10} 4.5 \), express it in terms of known logarithms. - Use the properties of logarithms to simplify the expression. - Substitute the known logarithm values and perform the calculations carefully. - The correct answer is \( \log_{10} 4.5 = 0.6532 \). ### Revision Points - Remember the properties of logarithms: \( \log(a/b) = \log a - \log b \) and \( \log(a^b) = b \cdot \log a \). - Always check your calculations step-by-step to avoid simple arithmetic errors. - Familiarize yourself with common logarithm values for quick reference. - Practice breaking down numbers into their prime factors to simplify logarithmic evaluations.