How to prove this logarithm equation? Given : log_(12)18 = a and log_(24)54=b prove that: ab+5(a−b)=1

Mark Rosales

Mark Rosales

Answered question

2022-11-12

How to prove this logarithm equation?
Given :
log 12 18 = a  and  log 24 54 = b
prove that:
a b + 5 ( a b ) = 1
My attempt: I couldn't solve it in any way, as base were not common. I could solve it if base of second equation was 12 raised to something, but here it is 12 × 2 :( Any help will be greatly appreciated.

Answer & Explanation

Eva Cochran

Eva Cochran

Beginner2022-11-13Added 14 answers

HINT:
a = log 12 18 = log ( 18 ) log ( 12 ) = log 2 + 2 log 3 2 log 2 + log 3
as log a b = log b log a and log ( a 2 b ) = log ( a 2 ) + log ( b ) = 2 log a + log b
Express log 2 in terms of log 3
Similarly for b
Then compare the values of log 2 to eliminate it

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