The question is as follows: If (log_ba)/(log_ca)=(19)/(99) then b/c=c^k. Compute k.

Awainaideannagi

Awainaideannagi

Answered question

2022-07-23

Logarithm Fraction Contest Math Question
The question is as follows:
If log b a log c a = 19 99 then b c = c k . Compute k

Answer & Explanation

juicilysv

juicilysv

Beginner2022-07-24Added 17 answers

First we convert all of the logarithms to base 10 (the base doesnt matter, they just should all be in the same base.) Then we cancel out the log a on the top and bottom and simplify the expression.
log b a log c a = log a log b log a log c = 1 log b 1 log c = log c log b = log b c = 19 99
This means that c = b 19 / 99 by definition. That means that we can replace c with b 19 / 99 in the formula b c
b c = b b 19 / 99 = b 99 / 99 b 19 / 99 = b 99 19 99 = b 80 99
Therefore we can take c k = b c and replace all of the parts and get
( b 19 99 ) k = ( b 19 k 99 ) = b 80 99
Since the base b is the same, then 19 k must equal 80, and k = 80 / 19
Nathalie Fields

Nathalie Fields

Beginner2022-07-25Added 2 answers

Rewrite
log b a log c a = 19 99 log a log b log a log c = 19 99 log a log b log c log a = 19 99 log c log b = 19 99 99 log c = 19 log b log c 99 = log b 19 (1) c 99 = b 19
and
(2) b c = c k b = c k + 1 b 19 = c 19 ( k + 1 ) .
Substituting ( 1 ) to ( 2 ) yields
c 99 = c 19 ( k + 1 ) 19 ( k + 1 ) = 99 k + 1 = 99 19 k = 99 19 1 = 80 19 .

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