Question

If \( \log_2 3 = p \), express \( \log_8 9 \) in terms of \( p \):

• A. \( \frac{2p}{3} \)
• B. \( \frac{3p}{2} \)
• C. \( \frac{4p}{3} \)
• D. \( \frac{p}{3} \)

Hint:

Use change of base: \( \log_b a = \frac{\log a}{\log b} \). Rewrite in terms of known logarithms.

Answer 1

The Correct Option is A

Step 1: Use change of base formula. \[ \log_8 9 = \frac{\log 9}{\log 8} = \frac{\log(3^2)}{\log(2^3)} = \frac{2\log 3}{3\log 2} \] Step 2: Express in terms of \( p \). \[ \log_2 3 = \frac{\log 3}{\log 2} = p \Rightarrow \frac{\log 3}{\log 2} = p \Rightarrow \log_8 9 = \frac{2}{3} \cdot p = \frac{2p}{3} \]