SOLUTION: solve the equation log(5x-1) + log(x+2)=1

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Question 313667: solve the equation log(5x-1) + log(x+2)=1
Found 2 solutions by nerdybill, Fombitz:
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
log(5x-1) + log(x+2)=1
log(5x-1)(x+2)=1
(5x-1)(x+2)=10^1
5x^2+10x-x-2 = 10
5x^2+9x-2 = 10
Use the quadratic formula to get our solutions:
x = {0.892, -2.692}
Checking for extraneous solutions we see that we can throw out the negative solution leaving:
x = 0.892
.
Details of quadratic follows:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 5x%5E2%2B9x%2B-12+=+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=%289%29%5E2-4%2A5%2A-12=321.

Discriminant d=321 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-9%2B-sqrt%28+321+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%289%29%2Bsqrt%28+321+%29%29%2F2%5C5+=+0.891647286716892
x%5B2%5D+=+%28-%289%29-sqrt%28+321+%29%29%2F2%5C5+=+-2.69164728671689

Quadratic expression 5x%5E2%2B9x%2B-12 can be factored:
5x%5E2%2B9x%2B-12+=+5%28x-0.891647286716892%29%2A%28x--2.69164728671689%29
Again, the answer is: 0.891647286716892, -2.69164728671689. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+5%2Ax%5E2%2B9%2Ax%2B-12+%29

Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
log%2810%2C%285x-1%29%29+%2B+log%2810%2C%28x%2B2%29%29=1
log%2810%2C%285x-1%29%28x%2B2%29%29=1
%285x-1%29%28x%2B2%29=10
5x%5E2%2B9x-2=10
5x%5E2%2B9x-12=0
Use the quadratic formula,
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-9+%2B-+sqrt%28+%28-9%29%5E2-4%2A5%2A%28-12%29+%29%29%2F%282%2A5%29+
x+=+%28-9+%2B-+sqrt%28+81%2B240%29%29%2F%2810%29+
x+=+%28-9+%2B-+sqrt%28+321%29%29%2F%2810%29+
Only the positive solution will be considered since the log function requires non-negative arguments.
x+=+%28-9+%2B+sqrt%28+321%29%29%2F%2810%29+ or approximately,
highlight%28x=+0.8916%29