If log_12 54=a then log_6 12=?

podvelkaj8

podvelkaj8

Answered question

2022-10-12

If log 12 54 = a then log 6 12 = ?
I am given
log 12 54 = a
So what will be value of log 6 12 ? I used base changing theorem and wrote expression as
log 6 54 log 6 12 = a
And then
1 + log 6 9 a = log 6 12
now what to do?

Answer & Explanation

Cavalascamq

Cavalascamq

Beginner2022-10-13Added 21 answers

Observe first that
a = log 12 ( 54 ) = log 12 ( 2 ) + 3 log 12 ( 3 ) = 1 log 2 ( 12 ) + 3 log 3 ( 12 ) = 1 2 + log 2 ( 3 ) + 3 1 + 2 log 2 ( 3 )
From this, we can compute log 2 ( 3 ) in terms of a. We just need to solve for x in the equation
a = 1 2 + x + 3 1 + 2 x
Then we can use this value to compute:
log 6 ( 12 ) = 2 log 6 ( 2 ) + log 6 ( 3 ) = 2 log 2 ( 6 ) + 1 log 3 ( 6 ) = 2 1 + log 2 ( 3 ) + 1 1 + 1 log 2 ( 3 )
The equation above is
a = 1 + 3 x x + 2
from where
x = 2 a 1 3 a
Therefore
log 6 ( 12 ) = 2 1 + 2 a 1 3 a + 1 1 + 1 2 a 1 3 a = 5 2 + a

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