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Sylvia Byrd

Sylvia Byrd

Answered question

2022-07-07

Prove that log 2 5 + log 2 7 > log 12
What I tried so far:
log 2 5 + log 2 7 > log 3 + log 4
( log 5 + log 7 ) 2 2 log 5 log 7 > log 3 + log 4
But it seems that I'm not even near the result.
Every suggestion / hint would be appreciated :)
Thanks in advance.
EDIT: log means log 10

Answer & Explanation

Leslie Rollins

Leslie Rollins

Beginner2022-07-08Added 25 answers

Without actually computing exact logs,...
log 10 5 log 10 5 + log 10 7 log 10 7 > log 10 12
log 10 5 log 10 7 + log 10 7 log 10 5 > log 10 12 log 10 5 log 10 7
Now LHS > 2 as it is the sum of a positive number ( 1) and its reciprocal. So it is sufficient to show that RHS < 2, which is equivalent to:
log 10 12 < 2 log 10 5 log 10 7 log 5 12 < log 10 49 3 log 5 12 < 3 log 10 49
But 12 3 = 1728 < 5 5 , while 49 3 > 10 5 shows 3 log 5 12 < 5 while 3 log 10 49 > 5
pipantasi4

pipantasi4

Beginner2022-07-09Added 6 answers

From 5 3 = 125 and 7 6 = 117649, we deduce that l o g ( 5 ) 2 3 and l o g ( 7 ) 5 6
From 3 ( 6 7 ) = 839808 and 5 9 = 1953125, we deduce that 3 ( 6 7 ) 5 9 and hence 12 8 10 9 . So l o g ( 12 ) 9 8 .
Finally, we have
l o g ( 5 ) 2 + l o g ( 7 ) 2 ( 2 3 ) 2 + ( 5 6 ) 2 = 41 36 = 82 72 81 72 9 8 l o g ( 12 )

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