To evaluate the expression \(log_{\sqrt{2}}4 + log_{\frac{1}{2}}16 - log_{4}32\), we will break it down step by step, using properties of logarithms and changing the base where necessary. ### Step 1: Evaluate \(log_{\sqrt{2}}4\) We can use the change of base formula for logarithms, which states that \(log_a b = \frac{log_c b}{log_c a}\). However, in this case, we can simplify directly since both the base and the argument can be expressed as powers of 2. 1. **Express \(\sqrt{2}\) and \(4\) in terms of base 2**: - \(\sqrt{2} = 2^{1/2}\) - \(4 = 2^2\) 2. **Use the change of base property**: \[ log_{\sqrt{2}}4 = log_{2^{1/2}}(2^2) \] 3. **Apply the logarithm power rule**: \[ log_{a^b} c = \frac{1}{b} log_a c \] Thus, \[ log_{2^{1/2}}(2^2) = \frac{1}{1/2} log_2(2^2) = 2 \cdot log_2(2^2) = 2 \cdot 2 = 4 \] ### Step 2: Evaluate \(log_{\frac{1}{2}}16\) 1. **Express \(\frac{1}{2}\) and \(16\) in terms of base 2**: - \(\frac{1}{2} = 2^{-1}\) - \(16 = 2^4\) 2. **Use the change of base property**: \[ log_{\frac{1}{2}}16 = log_{2^{-1}}(2^4) \] 3. **Apply the logarithm power rule**: \[ log_{2^{-1}}(2^4) = \frac{1}{-1} log_2(2^4) = -log_2(2^4) = -4 \] ### Step 3: Evaluate \(log_{4}32\) 1. **Express \(4\) and \(32\) in terms of base 2**: - \(4 = 2^2\) - \(32 = 2^5\) 2. **Use the change of base property**: \[ log_{4}32 = log_{2^2}(2^5) \] 3. **Apply the logarithm power rule**: \[ log_{2^2}(2^5) = \frac{1}{2} log_2(2^5) = \frac{1}{2} \cdot 5 = \frac{5}{2} \] ### Step 4: Combine the results Now we can substitute the evaluated logarithms back into the original expression: \[ log_{\sqrt{2}}4 + log_{\frac{1}{2}}16 - log_{4}32 = 4 - 4 - \frac{5}{2} \] ### Step 5: Simplify the expression 1. Combine the terms: \[ 4 - 4 = 0 \] Thus, we have: \[ 0 - \frac{5}{2} = -\frac{5}{2} \] ### Final Answer The final answer is: \[ -\frac{5}{2} = -2.5 \] ### Explanation of Options - **Option A (-5.5)**: This is incorrect because the calculations do not support such a negative value. - **Option B (-2.5)**: This is correct as shown in the calculations. - **Option C (2.5)**: This is incorrect because the result is negative, not positive. - **Option D (5.5)**: This is incorrect as it does not align with the evaluated logarithmic values. ### Revision Summary - Use properties of logarithms to simplify expressions. - Change the base of logarithms when necessary to a common base (like 2). - Remember the logarithm power rule: \(log_{a^b}(c) = \frac{1}{b} log_a(c)\). - Combine results carefully, paying attention to signs when adding and subtracting. This thorough breakdown should help you understand how to approach similar logarithmic problems in the future!