SOLUTION: Please help me solve {{{lnx+ln(x+3)=1}}}

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Question 419876: Please help me solve lnx%2Bln%28x%2B3%29=1
Found 2 solutions by nerdybill, josmiceli:
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
lnx%2Bln%28x%2B3%29=1
ln+x%28x%2B3%29=1
x%28x%2B3%29=+e%5E1
x%5E2%2B3x=+2.718
x%5E2%2B3x-2.718=+0
Solve by applying the quadratic formula to give:
x = {0.729, -3.729}
we can throw out the negative solution leaving:
x = 0.729
.
Details of quadratic formula follows:
.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B3x%2B-2.718281828+=+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=%283%29%5E2-4%2A1%2A-2.718281828=19.873127312.

Discriminant d=19.873127312 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-3%2B-sqrt%28+19.873127312+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%283%29%2Bsqrt%28+19.873127312+%29%29%2F2%5C1+=+0.728964294913671
x%5B2%5D+=+%28-%283%29-sqrt%28+19.873127312+%29%29%2F2%5C1+=+-3.72896429491367

Quadratic expression 1x%5E2%2B3x%2B-2.718281828 can be factored:
1x%5E2%2B3x%2B-2.718281828+=+1%28x-0.728964294913671%29%2A%28x--3.72896429491367%29
Again, the answer is: 0.728964294913671, -3.72896429491367. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B3%2Ax%2B-2.718281828+%29




Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
a general rule is:
+log%28a%2Ab%29+=+log%28a%29+%2B+log%28b%29+
For example:
+log+%2810%2A10%2A10%29+=+log%2810%29+%2B+log%2810%29+%2B+log%2810%29+
Just work the rule backwards
----------------------
+ln%28x%29+%2B+ln%28x+%2B+3%29+=+1+
+ln%28+x%2A%28x+%2B+3%29%29+=+1+
+ln%28+x%5E2+%2B+3x%29+=+1+
Now use the general rule
+log%28a%2Cb%29+=+c+ then,
+a%5Ec+=+b+
----------
+e%5E1+=+x%5E2+%2B+3x+
+x%5E2+%2B+3x+-+e+=+0
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
a+=+1
b+=+3
c+=-e
x+=+%28-3+%2B-+sqrt%28+3%5E2-4%2A1%2A%28-e%29+%29%29%2F%282%2A1%29+
x+=+%28-3+%2B-+sqrt%28+9+%2B+4e+%29%29%2F2+ answer