How solve this logarithms equation What relationship between a,b and c ? log a

sebadillab0

sebadillab0

Answered question

2022-07-09

How solve this logarithms equation
What relationship between a,b and c ?
log a u + log b u + log c u = log a b c u

Answer & Explanation

Savion Stanton

Savion Stanton

Beginner2022-07-10Added 10 answers

From Lucian's suggestion to transform log a u = ln u ln a etc. we find, using
x = ln a , y = ln b , z = ln c and cancelling the common log u factor, that
(1) 1 x + 1 y + 1 z 1 x + y + z = 0.
The left side factors into
(2) ( x + y ) ( x + z ) ( y + z ) x y z ( x + y + z ) .
Now since we are working with a , b , c being the base of logs, we are assuming each is positive and not 1, so that x , y , z are nonzero. We also know x + y + z 0 since it is the log of a b c which also appears as a logarithm base in the original equation.
So the top of ( 2 ) must be zero, which means some two of the three logs add to zero, which in turn means that at least one of the three possibilities a b = 1 , a c = 1 , b c = 1 must hold. It is then easy to check that on the other hand provided one of the three products is 1 then the statement holds for all u

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