Q26 / 58 · KVPY 2016 (SA) · Mathematics

Let $f$ be a function defined on the set of all positive integers such that $f(\mathrm{xy})=f(\mathrm{x})+f(\mathrm{y})$ for all positive integers $\mathrm{x}, \mathrm{y}$. If $f(12)=24$ and $f(8)=15$, the value of $f(48)$ is

  • A
    31
  • B
    32
  • C
    33
  • Correct answer
    D
    34

Solution

$f(x y)=f(x)+f(y)$ $\Rightarrow f(x)=\log _{a} x$ So, $f(12)=24$ $\Rightarrow \log _{a} 12=24$ $\Rightarrow 12=a^{24} \& f(8)=15$ $\Rightarrow \log _{a} 8=15$ $\Rightarrow 8=a^{15} \Rightarrow 2=a^{5}$ So, $f(48)=\log _{a} 48=\log _{a} 12+\log _{a} 4$ $=\log _{a} 12+\log _{a} 2^{2}$ $=24+2 \cdot 5$ $=34$

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

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