Question 841628
First Equation,
{{{log(x^2y)=a}}}
{{{2log(x)+log(y)=a}}}
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Second Equation,
{{{log(x/y)=b}}}
{{{log(x)-log(y)=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(x)=a+b}}}
{{{highlight(log(x)=(a+b)/3)}}}----Knowing this, find the formula for log(y).
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{{{log(x)-log(y)=b}}}
{{{log(x)-b=log(y)}}}
{{{log(y)=log(x)-b}}}
{{{log(y)=(a+b)/3-b}}} -----used substitution
{{{log(y)=((a+b)-3b)/3}}}
{{{highlight(log(y)=(a-2b)/3)}}}
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Almost done, ready to deal with the expression you want to transform.
SUMMARY:
{{{highlight(log(x)=(a+b)/3)}}} and {{{highlight(log(y)=(a-2b)/3)}}}
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The target expression to transform,
{{{highlight_green(log(y/x^2))}}}
{{{log(y)-2log(x)}}}
Now substitute what were found for log(x) and log(y).
{{{(a+b)/3-2(a-2b)/3}}}
{{{(a+b)/3-(2a-4b)/3}}}
{{{(a+b-2a+4b)/3}}}
{{{highlight(highlight((-a+5b)/3))}}}