ogarithmic inequality with different bases I have problem with following inequality log

boloman0z

boloman0z

Answered question

2022-06-20

ogarithmic inequality with different bases
I have problem with following inequality log 4 5 + log 5 6 + log 6 7 + log 7 8 4.4
I tried to change all logarithms to base 10 but it didn't work

Answer & Explanation

Donavan Mack

Donavan Mack

Beginner2022-06-21Added 24 answers

Rewrite the inequality as
1 4 ( log 5 log 4 + log 6 log 5 + log 7 log 6 + log 8 log 7 ) > 11 10
Now apply the AM-GM inequality:
1 4 ( log 5 log 4 + log 6 log 5 + log 7 log 6 + log 8 log 7 ) log 8 log 4 4 = 3 2 4
It remains to observe that
3 2 > ( 11 10 ) 4 = 14641 10000
hawatajwizp

hawatajwizp

Beginner2022-06-22Added 10 answers

You definitely have the right idea when it comes to a change of base. In general, let a , b , x R with a , b 1; a , b > 0 and x > 0. Then converting from base a to base b is done by the following:
log a x = log b x log b a
In your case, you want to convert each number to base 10 (so b = 10 ). In standard notation, log 10 is written as just log. Then your expression can be simplified like so:
log 4 5 + log 5 6 + log 6 7 + log 7 8 4.4 log 5 log 4 + log 6 log 5 + log 7 log 6 + log 8 log 7 4.4
and then see the answer from Barry Cipra.

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