Q268 / 270 · NDA 2016 (Phase 1) · Mathematics

If $\log _a(a b)=x$, then what is $\log _b(a b)$ equal to?

  • A
    $\frac{1}{x}$
  • B
    $\frac{x}{x+1}$
  • C
    $\frac{x}{1-x}$
  • Correct answer
    D
    $\frac{x}{x-1}$

Solution

$\begin{aligned} & \log _a a b=x \\ & \log _a a+\log _a b=x \\ & \frac{1}{\log _b a}=x-1\end{aligned}$ $\begin{aligned} \log _b a & =\frac{1}{x-1}...(i) \\ \log _b a b & =\log _b a+\log _b b \\ & =\frac{1}{x-1}+1 \quad(From (1))\\ & =\frac{1+x-1}{x-1} \\ \log _b a b & =\frac{x}{x-1}\end{aligned}$

Hence, the correct answer is option: 'D'.

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