SOLUTION: help me pleasee!! Solve simultaneously: {{{log xy =7}}} , {{{log (x/y)= 10}}} so far ... if {{{log (base 10) (x/y)=10}}} this implies that {{{(x/y)=10}}} therfore, {{{x=10y}

Algebra ->  Exponents -> SOLUTION: help me pleasee!! Solve simultaneously: {{{log xy =7}}} , {{{log (x/y)= 10}}} so far ... if {{{log (base 10) (x/y)=10}}} this implies that {{{(x/y)=10}}} therfore, {{{x=10y}      Log On


   

Question 97794This question is from textbook Introducing Pure Mathematics
: help me pleasee!! Solve simultaneously:
log+xy+=7 , log+%28x%2Fy%29=+10
so far ... if log+%28base+10%29+%28x%2Fy%29=10 this implies that %28x%2Fy%29=10
therfore, x=10y ... thats all i got! This question is from textbook Introducing Pure Mathematics

Found 3 solutions by jim_thompson5910, vincy_fyah, stanbon:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
log%2810%2C+%28xy%29%29+=7 , log+%2810%2C%28x%2Fy%29%29=+1 Start with the given system


xy=10%5E7 , x%2Fy=+10%5E1 Rewrite the logs using the property log%28b%2Cx%29=y <==> b%5Ey=x


xy=10000000 , x%2Fy=+10 Rewrite 10%5E7 as 10,000,000 for the first equation. Rewrite 10%5E1 as 10 for the second equation


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

%2810y%29y=10000000 Plug in x=10y into the first equation. This will eliminate x so we can solve for y


10y%5E2=10000000 Multiply


cross%2810%2F10%29y%5E2=10000000%2F10 Divide both sides by 10


y%2Ay=1000000 Divide


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



x=10%281000%29 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+%2810%2C%28xy%29%29+=7 Start with the first equation


log+%2810%2C%2810000%2A1000%29%29+=7 Plug in x=10000 and y=1000


log+%2810%2C%2810000000%29%29+=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+%2810%2C%28x%2Fy%29%29+=1 Start with the second equation


log+%2810%2C%2810000%2F1000%29%29+=1 Plug in x=10000 and y=1000


log+%2810%2C%2810%29%29+=1 Divide

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

Answer by vincy_fyah(4) About Me  (Show Source):
You can put this solution on YOUR website!
log+%28x%2Fy%29=1 .. sorry there...

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
log xy =7 , log (x/y)= 10
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logx + logy = 7
logx - logy = 10
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Add the two equations to get:
2logx = 17
log x = 8.5
x = 10^8.5
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Substitute that into one of the equations with y and solve for y:
log(10^8.5) + log y = 7
8.5 + logy = 7
logy = 7-8.5
logy = -1.5
y = 10^(-1.5)
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Cheers,
Stan H.