KVPY SA Exam 2016 Question Paper

Question : 15 of 80

Marks: +1, -0

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( 1 2 ) = 24

and

f( 8 ) = 15

, the value of

f( 4 8 )

Solution:

f (x y) = f (x) + f (y)

⇒ \f (\x) = \log_ax

So, f (12) = 24

⇒ \log_a 12 = 24

⇒ 12 = a^24

& f (8) = 15

⇒ \log_a 8 = 15

⇒ 8 = a^15 ⇒ 2 = a^5

So,

\f(48) = \log_a 48 = \log_a 12 + \log_a 4

= \log_a 12 + \log_a 2^2

= 24 + 2 . 5= 34

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