Question

If \( a = \log 2, b = \log 3, c = \log 4 \), then the value of \( \log(abcd) \) would be:

• A. \( \log_{10} 24 \)
• B. \( \log_2 24 \)
• C. \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{\log_5 4} \)
• D. \( \frac{1}{a} - \frac{1}{b} - \frac{1}{c} + \frac{1}{\log_4 5} \)

Hint:

Use \( \log a + \log b = \log(ab) \) and simplify all expressions before final computation.

Answer 1

The Correct Option is A

Given: \[ a = \log 2,\quad b = \log 3,\quad c = \log 4 = \log (2^2) = 2\log 2 = 2a \] Let’s compute: \[ \log(abcd) = \log a + \log b + \log c + \log d \] Assuming \( d = \log 1 \) is implied or undefined. If \( d = \log 1 = 0 \), discard. But in the context of the question, probably asking: \[ \log(abcd) = \log(\log 2 \cdot \log 3 \cdot \log 4) \Rightarrow a + b + c = \log 2 + \log 3 + \log 4 = \log(2 \cdot 3 \cdot 4) = \log 24 \] \[ {\log 24} \]