SOLUTION: logarithem silmautanious equation. log base8 (xy)=3, (log base2 x)(log base2 y)=18

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Question 977136: logarithem silmautanious equation.
log base8 (xy)=3,
(log base2 x)(log base2 y)=18
Found 2 solutions by josgarithmetic, anand429:
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
This is not going to be yet the complete solution, but is just part of the steps to get to it:

First equation is log%288%28xy%29%29=3

log%282%2C%28xy%29%29%2Flog%282%2C8%29=3

log%282%2C%28xy%29%29%2Flog%282%2C%282%5E3%29%29=3

log%282%2C%28xy%29%29%2F3=3

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log%282%2C%28xy%29%29=9-----In this form, the equation will be easier to relate to the other given equation.

You should work through that on your own including the second given equation in the system of equations.
You should be able to form through the steps something like,
%28log%282%2Cx%29-3%29%28log%282%2Cx%29-6%29=0
which means that either log%282%2Cx%29=3 OR log%282%2Cx%29=6.

The values are turned around for dealing logs of y.

You should then be able to finish find x and y.
One of them is 2^3 and the other is 2^6.

Answer by anand429(138) About Me  (Show Source):
You can put this solution on YOUR website!
log%288%2Cxy%29=3
=> 8%5E3+=+xy ----(i)
Also,
log%282%2Cx%29+%2A+log+%282%2Cy%29+=18
=>log%282%2Cx%29%2Alog%282%2C512%2Fx%29=18 (using (i))
=> log%282%2Cx%29%2A%28log%282%2C512%29-log%282%2Cx%29%29=18
=> log%282%2Cx%29%2A%28log%282%2C2%5E9%29-log%282%2Cx%29%29=18
=> log%282%2Cx%29%2A%289%2Alog%282%2C2%29-log%282%2Cx%29%29=18
=> log%282%2Cx%29%2A%289%2A1-log%282%2Cx%29%29=18
Taking log%282%2Cx%29 as p,
p%2A%289-p%29=18
=> p%5E2-9p%2B18+=+0
=> %28p-3%29%28p-6%29=0
=> +p=3 or p=6
So,
log%282%2Cx%29+=+3 or log%282%2Cx%29+=+6
=> x=2%5E3 or x=2%5E6
=> x=8 or x=64
=> y=64 or y=8 (Using eqn. (i))
So the answer sets of x and y are (8,64) or (64,8)