SOLUTION: Solve the equation. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) log(x + 3) = 1 − log(x − 2) Since the bases are the same I s

Algebra ->  Equations -> SOLUTION: Solve the equation. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) log(x + 3) = 1 − log(x − 2) Since the bases are the same I s      Log On


   

Question 1156594: Solve the equation. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)
log(x + 3) = 1 − log(x − 2)
Since the bases are the same I set them equal to each other and got x+3=1-x+2 and solved to get x=0 which is incorrect. What did I do wrong?
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Solve the equation. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)
log(x + 3) = 1 − log(x − 2)
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Do NOT enter (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) What is the point of that?
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log(x + 3) = 1 − log(x − 2)

Assuming base 10:
log(x + 3) = log(10) − log(x − 2)
log(x+3) = log(10/(x-2))
x+3 = 10/(x-2)
x^2 + x - 6 = 10
x^2 + x - 16 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B1x%2B-16+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%281%29%5E2-4%2A1%2A-16=65.

Discriminant d=65 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-1%2B-sqrt%28+65+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%281%29%2Bsqrt%28+65+%29%29%2F2%5C1+=+3.53112887414927
x%5B2%5D+=+%28-%281%29-sqrt%28+65+%29%29%2F2%5C1+=+-4.53112887414927

Quadratic expression 1x%5E2%2B1x%2B-16 can be factored:
1x%5E2%2B1x%2B-16+=+%28x-3.53112887414927%29%2A%28x--4.53112887414927%29
Again, the answer is: 3.53112887414927, -4.53112887414927. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B1%2Ax%2B-16+%29

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The negative solution gives a negative argument for the log, ignore it.
x =~ 3.5311
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