Question

If \(\log_a(ab) = x\) then the value of \(\log_b(ab)\) is:

• A. \(\fracxx+1\)
• B. \(\fracxx-1\)
• C. \(\fracx1-x\)
• D. \(\frac1x\)

Hint:

Use base-change relationships: \(\log_a b = \frac1\log_b a\), and break down products using \(\log(ab) = \log a + \log b\).

Answer 1

The Correct Option is B

Given: \(\log_a(ab) = x\).
Using log properties: \(\log_a(ab) = \log_a a + \log_a b = 1 + \log_a b\).
So: \(1 + \log_a b = x \Rightarrow \log_a b = x - 1\).
We want \(\log_b(ab)\):
\(\log_b(ab) = \log_b a + \log_b b = \log_b a + 1\).
From \(\log_a b = x - 1\), we have \(\log_b a = \frac1\log_a b = \frac1x - 1\).
Thus: \(\log_b(ab) = \frac1x - 1 + 1 = \frac1 + (x - 1)x - 1 = \fracxx - 1\).
Hence, the correct answer is \(\fracxx - 1\).