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Grade: 12th pass

                        

let f(x)= log((1-x)/(1+x)), find x,y for which f(x)+f(y)=f((x+y)/(1+xy))

2 years ago

Answers : (1)

Arun
24741 Points
							
Dear Aman
 
f(x) = ln (1-x/1+x)
Given that f(x) +f (y) = f(x+y /1 +xy)
ln (1-x/1+x) + ln (1-y/1+y) = ln (1 -(x+y)/1 +x+y)
ln((1-x)*(1-y)/(1+x)*(1+y)) = ln (1-(x+y)/1+x+y)
Removing log
And. Then on solving
We get
2xy(x+y) = 0
Hence xy = 0  or x+y = 0
For xy = 0, solution is but obvious
For x+y = 0, x = -y
Putting in the given equation
ln((1+x)*(1-x)/(1-x)(1+x)) = ln1
LHS and RHS contradicts
Hence 
x = 0 = y
Also
1-x >0
x 1
hence
x belongs to (0, 1)
 
Regards
Arun (askIITians forum expert)
2 years ago
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