Question

which is equivalent to \\(\log_{\frac{1}{2}} 462\\)?

• \\(\frac{\log_{\frac{1}{2}} 462}{\log_{462} \frac{1}{2}}\\)
• \\(\frac{\log 462}{\log \frac{1}{2}}\\)
• \\(\frac{\log \frac{1}{2}}{\log 462}\\)

Explanation:

Identify the given expression

Using the Logarithmic Evaluation knowledge point

$$ \log_{\frac{1}{2}} 462 $$

Apply the change of base formula

The change of base formula states that for any positive bases \(a, b\) and argument \(x\):

$$ \log_a x = \frac{\log_b x}{\log_b a} $$

Using common logarithms (base 10, written as \(\log\)):

$$ \log_{\frac{1}{2}} 462 = \frac{\log 462}{\log \frac{1}{2}} $$

Match with the given options

We compare our derived expression with the choices:

• Option 1: \(\frac{\log_{\frac{1}{2}} 462}{\log_{462} \frac{1}{2}}\)
• Option 2: \(\frac{\log 462}{\log \frac{1}{2}}\)
• Option 3: \(\frac{\log \frac{1}{2}}{\log 462}\)

The second option matches our result.

Answer:

• (A) \(\frac{\log_{\frac{1}{2}} 462}{\log_{462} \frac{1}{2}}\)
• (B) \(\frac{\log 462}{\log \frac{1}{2}}\) (Correct answer)
• (C) \(\frac{\log \frac{1}{2}}{\log 462}\)