Search any question & find its solution
Question:
Answered & Verified by Expert
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
Options:
Solution:
1664 Upvotes
Verified Answer
The correct answer is:
34
$f(x y)=f(x)+f(y)$
$\Rightarrow f(x)=\log _{a} x$
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, $f(48)=\log _{a} 48=\log _{a} 12+\log _{a} 4$
$=\log _{a} 12+\log _{a} 2^{2}$
$=24+2 \cdot 5$
$=34$
$\Rightarrow f(x)=\log _{a} x$
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, $f(48)=\log _{a} 48=\log _{a} 12+\log _{a} 4$
$=\log _{a} 12+\log _{a} 2^{2}$
$=24+2 \cdot 5$
$=34$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.