How to prove that: log_(1/2)(3) + log_3(1/2) < -2 Please help me solve it.

snaketao0g

snaketao0g

Answered question

2022-10-31

How to prove that: log 1 2 ( 3 ) + log 3 ( 1 2 ) < 2
Prove that:
log 1 2 ( 3 ) + log 3 ( 1 2 ) < 2
Please help me solve it.

Answer & Explanation

Bridget Acevedo

Bridget Acevedo

Beginner2022-11-01Added 19 answers

If you know the AM-GM inequality, this is easy.
a = log 1 2 ( 3 ) = log 2 3 and b = log 3 ( 1 2 ) = log 3 2.
Now a > 0 , b > 0 so a b > 2 | a b | = 2 implying a + b < 2
Rene Nicholson

Rene Nicholson

Beginner2022-11-02Added 2 answers

Let a = log 3 1 2 , so a < 0 and log 1 2 3 = log 3 3 log 3 1 2 = 1 a .
Then ( a + 1 ) 2 > 0 a 2 + 1 > 2 a a + 1 a < 2 (since a < 0)

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