SOLUTION: Show that {{{(log(9,(x/y)))=((1/2)(log(3,x)-log(3,y)))}}}. Hence, find the values of {{{x}}} and {{{y}}} which satisfies the equation {{{(log(9,(x/y)))=(1/4)}}} and {{{(3(log(3,x))

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Show that {{{(log(9,(x/y)))=((1/2)(log(3,x)-log(3,y)))}}}. Hence, find the values of {{{x}}} and {{{y}}} which satisfies the equation {{{(log(9,(x/y)))=(1/4)}}} and {{{(3(log(3,x))      Log On


   

Question 763181: Show that %28log%289%2C%28x%2Fy%29%29%29=%28%281%2F2%29%28log%283%2Cx%29-log%283%2Cy%29%29%29. Hence, find the values of x and y which satisfies the equation %28log%289%2C%28x%2Fy%29%29%29=%281%2F4%29 and %283%28log%283%2Cx%29%29%29=%282%28log%283%2Cy%29%29%29.
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Show that log%289%2C%28x%2Fy%29%29=%281%2F2%29%28log%283%2Cx%29-log%283%2Cy%29%29
log%283%2Cx%2Fy%29+=+log%289%2Cx%2Fy%29%2Flog%283%2C9%29+=+log%289%2Cx%2Fy%29%2F2
log%28b%2Cx%2Fy%29+=+log%28b%2Cx%29+-+log%28b%2Cy%29 b = any base
====================================================
Hence, find the values of x and y which satisfies the equation
log%289%2Cx%2Fy%29=1%2F4 and 3log%283%2Cx%29=2log%283%2Cy%29
log%289%2Cx%2Fy%29=1%2F4 --> log%283%2Cx%2Fy%29=1%2F2
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log%283%2Cx%2Fy%29=1%2F2
3log%283%2Cx%29=2log%283%2Cy%29 --> log%283%2Cx%5E3%29=log%283%2Cy%5E2%29
x%5E3+=+y%5E2
y+=+x%5E%281.5%29 (fractional exponents don't display well)
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log%283%2Cx%2Fy%29=1%2F2
log%283%2Cx%2Fx%5E1.5%29=1%2F2
log%283%2Cx%5E%28-0.5%29%29+=+1%2F2
1%2Fsqrt%28x%29+=+sqrt%283%29
x = 1/3
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y = x^1.5
y+=+sqrt%283%29%2F9