Question 936231: log base 9 of 6=a, log base 27 of 18=b. Express b in terms of a.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! log(9,6) = a
log(27,18) = b
you want to express b in terms of a.
log(9,6) = a if and only if 9^a = 6
log(27,18) = b if and only if 27^b = 18
since 9 = 3^2, then 9^a can be expressed as (3^2)^a which becomes 3^(2a) and you get:
9^a = 6 becomes 3^(2a) = 6
since 27 = 3^3, then 27^b can be expressed as (3^3)^b which becomes 3^(3b) and you get:
27^b = 18 becomes 3^(3b) = 18
if you divide both sides of this equation by 3, you will get:
3^(3b) / 3 = 6
simplify to get:
3^(3b-1) = 6
you now have:
3^(2a) = 6 and 3^(3b-1) = 6
this can only be true if 2a = 3b-1
solve for b and you get b = (2a+1) / 3
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