Don't understand simple logarithm problem with fractional base 3 log <mrow class=

Izabella Ponce

Izabella Ponce

Answered question

2022-06-19

Don't understand simple logarithm problem with fractional base
3 log 4 9 27 8 4
log 3 2 16 81
I understand using the expansion property to expand the division into a subtraction but how do I proceed from there?

Answer & Explanation

Hadley Cunningham

Hadley Cunningham

Beginner2022-06-20Added 20 answers

First observe that
3 log 4 9 27 8 4 = 3 4 log 4 9 27 8
So all we need now is to compute the log term. By definition log a b = y means that a y = b so we need to find y such that
( 4 9 ) y = 27 8
Expanding this, we have
( 2 2 3 2 ) y = 3 3 2 3
or
( 2 3 ) 2 y = ( 3 2 ) 3
and we're almost there, if we could make the left and right sides have the same base. Can you carry on from there?
Kyla Ayers

Kyla Ayers

Beginner2022-06-21Added 8 answers

The trick here is to not go to the subtraction step. Observe, for example, for the second expression that 16 81 = ( 2 3 ) 4 = ( 3 2 ) 4 , and hence the second logarithm is equal to 4. The first one is similar. Let me know if you have any difficulties.

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