Infinite inequality with logarithm Solve inequality 3 -log_(0.5)x - log^2_(0.5)x - log^3_(0.5)x - ... >= 4 log_(0.5)x Any suggestions how to start?

bruinhemd3ji

bruinhemd3ji

Answered question

2022-11-17

Infinite inequality with logarithm
Solve inequality 3 log 0.5 x log 0.5 2 x log 0.5 3 x 4 log 0.5 x Any suggestions how to start?

Answer & Explanation

Abril Orr

Abril Orr

Beginner2022-11-18Added 14 answers

We have
3 log 0.5 x log 0.5 2 x log 0.5 3 x = 4 n = 0 log 0.5 n x = 4 1 1 log 0.5 x = 3 4 log 0.5 x 1 log 0.5 x
for | log 0.5 x | < 1 0.5 < x < 2. Now note that the inequality
3 4 log 0.5 x 1 log 0.5 x 4 log 0.5 x
is equivalent to
( 2 log 0.5 x 3 ) ( 2 log 0.5 x 1 ) 0
which implies log 0.5 x 0.5 or x 2 / 2. Therefore, we have
2 / 2 x < 2

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