Prove that log 2 </msup> &#x2061;<!-- ⁡ --> 5 + log 2 </msup> &#

Sylvia Byrd 2022-07-07 Answered
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
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (2)

Leslie Rollins
Answered 2022-07-08 Author has 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

We have step-by-step solutions for your answer!

pipantasi4
Answered 2022-07-09 Author has 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 )

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

New questions