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Bruno Pittman 2022-07-09 Answered
If a + b 1 = 1 + l n ( 2 a 1 ) l n 4 + l n ( 2 b 1 ) l n 4 then a = b?
where a , b > 0 are real numbers and ln is l o g e , then is a=b?
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Answers (1)

furniranizq
Answered 2022-07-10 Author has 20 answers
Let f ( x ) = x log 4 ( 2 x + 2 4 ) . Then the given relation is f ( a ) + f ( b ) = 0
Now f ( x ) has a minimum at x = 1, which is f ( 1 ) = 0. So f ( x ) > 0 for all other values of x > 0. Hence for the relation to be satisfied, a = b = 1
Updated: to show the minimum without calculus, note that
f ( x ) 0 x log 4 ( 2 x + 2 4 ) 4 x 2 2 + x 4 y 2 4 y 4 ( y 2 ) 2 0
where y = 2 x
and this also f ( x ) = 0 2 x = 2 x = 1

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