Question

1. multiple choice $\log_{1/2} x^2 =$ (a) $-2 \log_2 x$ (b) $2 \log_2 x$ (c) $-0.5 \log_2 x$ (d) $0.5 \log_2 x$ (e) $-2 \log_2 |x|$

Explanation:

Step1: Apply the logarithm power rule

The power rule of logarithms states that \(\log_a M^n=n\log_a M\). So, \(\log_{1/2}x^{2}=2\log_{1/2}|x|\) (we use \(|x|\) because \(x^{2}\) is non - negative and the domain of \(\log\) function requires the argument to be positive).

Step2: Use the change of base formula

The change of base formula is \(\log_a b=\frac{\log_c b}{\log_c a}\). Here, \(a = \frac{1}{2}\), \(b = |x|\), and \(c = 2\). So, \(\log_{1/2}|x|=\frac{\log_2|x|}{\log_2\frac{1}{2}}\).
Since \(\log_2\frac{1}{2}=\log_22^{-1}=- 1\) (using the rule \(\log_a a^k = k\)).
Then \(\log_{1/2}|x|=-\log_2|x|\).

Step3: Substitute back

Substitute \(\log_{1/2}|x|=-\log_2|x|\) into \(2\log_{1/2}|x|\). We get \(2\log_{1/2}|x|=2\times(-\log_2|x|)=-2\log_2|x|\).

Answer:

E. \(-2\log_2|x|\)