Question

A function defined for positive real numbers and satisfying f(xy) = f(x) + f(y) ; f(2) = 1, then the value of f(8) is equal to


Zero


1


2


3

Answer 1

Answer:

f ( 8 ) = 3

Step-by-step explanation:

Given :

f ( x y ) = f ( x ) + f ( y ) , ∀ x , y ∈ R and f ( 2 ) = 1

We are asked to find f ( 8 ) :

We know :

If , f ( x y ) = f ( x ) + f ( y )

= > f ( x ) = k ㏑ ( x )

f ( 2 ) = k ㏑ ( 2 ) = 1

= > k = 1 / ㏑2

Now :

f ( 8 ) = k ㏑ 8

= > k ㏑ 2³

= > 3 k ㏑ 2

Putting value of k = 1 / ㏑ 2 we get :

f ( 8 ) = 3 ㏑ 2 / ㏑ 2

= > f ( 8 ) = 3

Therefore , we get required answer!

Answer 2

Step-by-step explanation:

$\bf \underline{Question}$

A function defined for positive real numbers and satisfying f(xy) = f(x) + f(y) ; f(2) = 1, then the value of f(8) is equal to

Zero

1

2

3

_________________________________

$\bf \underline{Given}$

• f(xy) = f(x) + f(y) ;

• f(2) = 1,

_________________________________

$\bf \underline{To\:Find}$

• What is the value of of f(8)

_________________________________

_________________________________$\bf \underline{</u></strong><strong><u>solve</u></strong><strong><u>\</u></strong><strong><u>t</u></strong><strong><u>o</u></strong><strong><u>}$

According to the question:-

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

f ( x ) = M s ( x )

Given in the question:-

f ( 2 ) = M. s ( 2 ) = 1

Then

$\sf \: M = \frac{1}{s \: 2}$

According to the task

we have

f ( 8 ) = M s 8

According to the lcm :-

$\sf \to \: 8 = 2 \times 2 \times 2 \\ </p><p></p><p> \sf \: Then \\ \sf \to \: 2^3 \\ </p><p> </p><p> \sf \to \: M .. s 2^3</p><p></p><p> \sf \to \: 3 M. .s 2$

Putting all values in question

$\sf \: f ( 8 ) = \frac{3 \: {s}^{2} }{ {s}^{2} }$

$\sf \blue{f ( 8 ) = 3. }$