SOLUTION: If {{{lg x^2y=a}}} and {{{lg (x/y) = b}}}, express {{{lg (y/x^2)}}} in terms of a and b.

Algebra ->  Exponents -> SOLUTION: If {{{lg x^2y=a}}} and {{{lg (x/y) = b}}}, express {{{lg (y/x^2)}}} in terms of a and b.      Log On


   

Question 841628: If lg+x%5E2y=a and lg+%28x%2Fy%29+=+b, express lg+%28y%2Fx%5E2%29 in terms of a and b.
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
First Equation,
log%28x%5E2y%29=a
2log%28x%29%2Blog%28y%29=a
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Second Equation,
log%28x%2Fy%29=b
log%28x%29-log%28y%29=b
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Those give a system of equations in unknowns log(x) and log(y), which can be solved.
SYSTEM:
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2log(x)+log(y)=a
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log(x)-log(y)=b
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Elimination Method will be most efficient.
Add left members and add right members.
3log%28x%29=a%2Bb
highlight%28log%28x%29=%28a%2Bb%29%2F3%29----Knowing this, find the formula for log(y).
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log%28x%29-log%28y%29=b
log%28x%29-b=log%28y%29
log%28y%29=log%28x%29-b
log%28y%29=%28a%2Bb%29%2F3-b -----used substitution
log%28y%29=%28%28a%2Bb%29-3b%29%2F3
highlight%28log%28y%29=%28a-2b%29%2F3%29
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Almost done, ready to deal with the expression you want to transform.
SUMMARY:
highlight%28log%28x%29=%28a%2Bb%29%2F3%29 and highlight%28log%28y%29=%28a-2b%29%2F3%29
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The target expression to transform,
highlight_green%28log%28y%2Fx%5E2%29%29
log%28y%29-2log%28x%29
Now substitute what were found for log(x) and log(y).
%28a%2Bb%29%2F3-2%28a-2b%29%2F3
%28a%2Bb%29%2F3-%282a-4b%29%2F3
%28a%2Bb-2a%2B4b%29%2F3
highlight%28highlight%28%28-a%2B5b%29%2F3%29%29