Question 97794
{{{log(10, (xy)) =7}}} , {{{log (10,(x/y))= 1}}} Start with the given system



{{{xy=10^7}}} , {{{x/y= 10^1}}} Rewrite the logs using the property {{{log(b,x)=y}}} <==> {{{b^y=x}}}



{{{xy=10000000}}} , {{{x/y= 10}}} Rewrite {{{10^7}}} as 10,000,000 for the first equation. Rewrite {{{10^1}}} as 10 for the second equation



{{{x=10y}}} Solve the second equation for x by multiplying both sides by y


{{{(10y)y=10000000}}} Plug in {{{x=10y}}} into the first equation. This will eliminate x so we can solve for y



{{{10y^2=10000000}}} Multiply



{{{cross(10/10)y^2=10000000/10}}} Divide both sides by 10



{{{y*y=1000000}}} Divide



{{{y=1000}}} Take the square root of both sides. So this is our first answer




{{{x=10(1000)}}} Plug in {{{y=1000}}} into {{{x=10y}}} to find x



{{{x=10000}}} Multiply



So our answer is 

{{{x=10000}}} and {{{y=1000}}}



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Check:


{{{log (10,(xy)) =7}}} Start with the first equation



{{{log (10,(10000*1000)) =7}}} Plug in {{{x=10000}}} and {{{y=1000}}}



{{{log (10,(10000000)) =1}}} Multiply



{{{7 =7}}} Evaluate the left side. So the solutions {{{x=10000}}} and {{{y=1000}}} make the first equation true


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{{{log (10,(x/y)) =1}}} Start with the second equation



{{{log (10,(10000/1000)) =1}}} Plug in {{{x=10000}}} and {{{y=1000}}}



{{{log (10,(10)) =1}}} Divide


{{{1 =1}}} Evaluate the left side. So the solutions {{{x=10000}}} and {{{y=1000}}} make the second equation true