SOLUTION: I need help solving this logarithmic equation: log(x-1)+log(x+2)=2

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Question 145525: I need help solving this logarithmic equation:
log(x-1)+log(x+2)=2
Found 2 solutions by jim_thompson5910, stanbon:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
log%2810%2C%28x-1%29%29%2Blog%2810%2C%28x%2B2%29%29=2 Start with the given equation


log%2810%2C%28%28x-1%29%28x%2B2%29%29%29=2 Combine the logs using the identity log%28b%2C%28A%29%29%2Blog%28b%2C%28B%29%29=log%28b%2C%28A%2AB%29%29



log%2810%2C%28x%5E2%2Bx-2%29%29=2 Foil


10%5E2=x%5E2%2Bx-2 Rewrite the equation using the property: log%28b%2C%28x%29%29=y ====> b%5Ey=x


100=x%5E2%2Bx-2 Square 10 to get 100


0=x%5E2%2Bx-102 Subtract 100 from both sides


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%281%29+%2B-+sqrt%28+%281%29%5E2-4%281%29%28-102%29+%29%29%2F%282%281%29%29 Plug in a=1, b=1, and c=-102


x+=+%28-1+%2B-+sqrt%28+1-4%281%29%28-102%29+%29%29%2F%282%281%29%29 Square 1 to get 1.


x+=+%28-1+%2B-+sqrt%28+1--408+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-102%29 to get -408


x+=+%28-1+%2B-+sqrt%28+1%2B408+%29%29%2F%282%281%29%29 Rewrite sqrt%281--408%29 as sqrt%281%2B408%29


x+=+%28-1+%2B-+sqrt%28+409+%29%29%2F%282%281%29%29 Add 1 to 408 to get 409


x+=+%28-1+%2B-+sqrt%28+409+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%28-1%2Bsqrt%28409%29%29%2F%282%29 or x+=+%28-1-sqrt%28409%29%29%2F%282%29 Break up the expression.


So the possible answers are x+=+%28-1%2Bsqrt%28409%29%29%2F%282%29 or x+=+%28-1-sqrt%28409%29%29%2F%282%29

However, since you cannot take the log of a negative number, this means that the only answer is

x+=+%28-1%2Bsqrt%28409%29%29%2F%282%29



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Answer:


So our answer is x+=+%28-1%2Bsqrt%28409%29%29%2F%282%29 which approximates to x=9.612

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
log(x-1)+log(x+2)=2
log[x^2+x-2] = 2
x^2+x-2 = 10^2
x^2+x-102 = 0
------
x = [-1 +- sqrt(1-4*1*-102)]/2
x = [-1 +- sqrt(409)]/2
x = [-1 +- 20.22]2
Positive solution:
x = 9.612
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Cheers,
Stan H.[