Michael was asked to properly evaluate this logarithm. Which answer should he come to? \( \log _{9} 1 / 2 \) - \( -1 / 2 \) -9

To evaluate the logarithm \( \log_{9} \left( \frac{1}{2} \right) \), we need to determine the exponent \( x \) such that: \[ 9^x = \frac{1}{2} \] Let's analyze the given options: 1. **Option 1: \( x = -\frac{1}{2} \)** Calculate \( 9^{-1/2} \): \[ 9^{-1/2} = \frac{1}{\sqrt{9}} = \frac{1}{3} \] This does not equal \( \frac{1}{2} \). 2. **Option 2: \( x = -9 \)** Calculate \( 9^{-9} \): \[ 9^{-9} = \frac{1}{9^9} \] This is an extremely small number, far less than \( \frac{1}{2} \). Neither of the provided options exactly equals \( \log_{9} \left( \frac{1}{2} \right) \). However, if we were to approximate or consider the closest option, \( -\frac{1}{2} \) is closer in magnitude to the actual value compared to \( -9 \). **Actual Calculation:** For a more precise value, we can use the change of base formula: \[ \log_{9} \left( \frac{1}{2} \right) = \frac{\log \left( \frac{1}{2} \right)}{\log 9} = \frac{-\log 2}{2\log 3} \approx -0.315 \] This value is closer to \( -\frac{1}{2} \) than to \( -9 \). **Conclusion:** Given the options, **\( -\frac{1}{2} \)** is the more appropriate choice. **Answer:** Michael should choose the answer \( -\frac{1}{2} \).