A functional question

Algebra Level 2

Let f ( x ) f(x) be a real-valued function defined for all x > 0 x > 0 such that f ( a b ) = f ( a ) + f ( b ) f(ab) = f(a) + f(b) for all a , b > 0 a, b > 0 .

What is the value of f ( 1 ) f(1) ?

0.5 1 Insufficient information provided to answer 0

Denton Young by Denton Young , 47, USA

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3 solutions

Denton Young
Feb 13, 2018

f ( 1 × a ) f(1 \times a) = f ( 1 ) + f ( a ) f(1) + f(a)

f ( a ) = f ( 1 ) + f ( a ) f(a) = f(1) + f(a)

f ( a ) − f ( a ) = f ( 1 ) f(a) - f(a) = f(1)

0 = f ( 1 ) 0 = f(1)

2 Helpful 0 Interesting 0 Brilliant 0 Confused

f ( 1 ) = f ( 1 ⋅ 1 ) = f ( 1 ) + f ( 1 ) = 2 f ( 1 ) ∴ f ( 1 ) = 0. f(1) = f(1 \cdot 1) = f(1) + f(1) = 2f(1) \;\;\; \therefore f(1) = 0.

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Mahmoud Khattab
Feb 14, 2018

The only function that meets the condition is the logarithmic function, accordingly Log ( 1 ) = 0

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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