Use the change-of-base formula and a calculator to evaluate the logarithm. Round your answer to three decimal places. \( \log _{1 / 2} 6 \) \( \log _{1 / 2} 6 \approx \square \) (Do not round until the final answer. Then round to three decimal places as needed.)

To evaluate the logarithm \( \log_{1/2} 6 \) using the change-of-base formula, we can use the formula: \[ \log_b a = \frac{\log_c a}{\log_c b} \] where \( a \), \( b \), and \( c \) are positive real numbers and \( c \neq 1 \). In this case, we have \( a = 6 \), \( b = \frac{1}{2} \), and \( c = 10 \) (since we are using a calculator to evaluate the logarithm). Substituting these values into the change-of-base formula, we get: \[ \log_{1/2} 6 = \frac{\log_{10} 6}{\log_{10} \frac{1}{2}} \] Now, we can use a calculator to evaluate the logarithms and find the approximate value of \( \log_{1/2} 6 \). Let's calculate this using a calculator. Calculate the value by following steps: - step0: Calculate: \(\frac{\log_{10}{\left(6\right)}}{\log_{10}{\left(\frac{1}{2}\right)}}\) - step1: Simplify: \(\frac{\log_{10}{\left(6\right)}}{-\log_{10}{\left(2\right)}}\) - step2: Use the logarithm base change rule: \(-\log_{2}{\left(6\right)}\) - step3: Rewrite the expression: \(-1-\log_{2}{\left(3\right)}\) The approximate value of \( \log_{1/2} 6 \) is approximately -2.584963. Rounding this to three decimal places, we get: \[ \log_{1/2} 6 \approx -2.585 \]