(D) Given the functional equation $f(xy) = f(x) + f(y)$.We know $f(8) = f(2 \cdot 2 \cdot 2) = f(2) + f(2) + f(2) = 3f(2)$.Given $f(8) = 15$,so $3f(2)

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(D) Given the functional equation $f(xy) = f(x) + f(y)$.We know $f(8) = f(2 \cdot 2 \cdot 2) = f(2) + f(2) + f(2) = 3f(2)$.Given $f(8) = 15$,so $3f(2) = 15$,which implies $f(2) = 5$.Now,we need to find $f(48)$.We can write $48 = 12 \cdot 4 = 12 \cdot 2 \cdot 2$.Using the property $f(xy) = f(x) + f(y)$,we get:$f(48) = f(12 \cdot 2 \cdot 2) = f(12) + f(2) + f(2)$.Substituting the known values $f(12) = 24$ and $f(2) = 5$:$f(48) = 24 + 5 + 5 = 34$.Thus,the correct option is $D$.

Class 12 Mathematics Relation and Function Functional Equations

Difficult

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

KVPY 2016 All KVPY

A
$31$
B
$32$
C
$33$
D
$34$

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

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