SOLUTION: how do you solve log x + log (3x + 5)= 1

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Question 745948: how do you solve log x + log (3x + 5)= 1
Found 2 solutions by savvyhush23, nerdybill:
Answer by savvyhush23(50) About Me  (Show Source):
You can put this solution on YOUR website!
how do you solve log x + log (3x + 5)= 1
Properties of Logarithm:
log M + log N = log MN
log (10) = 1
So, +log+%28%28x%29%29+%2B+log+%28%283x%2B5%29%29+=+1
log%28%28x%283x%2B5%29%29%29+=+log+%28%2810%29%29
log+%28%283x%5E2%2B5x%29%29+=+log+%28%2810%29%29 eliminate log
3x%5E2%2B5x+=+10
3x%5E2%2B5x+-10=0,it is a quadratic equation,
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B5x%2B-10+=+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=%285%29%5E2-4%2A3%2A-10=145.

Discriminant d=145 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-5%2B-sqrt%28+145+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%285%29%2Bsqrt%28+145+%29%29%2F2%5C3+=+1.17359909646538
x%5B2%5D+=+%28-%285%29-sqrt%28+145+%29%29%2F2%5C3+=+-2.84026576313205

Quadratic expression 3x%5E2%2B5x%2B-10 can be factored:
3x%5E2%2B5x%2B-10+=+3%28x-1.17359909646538%29%2A%28x--2.84026576313205%29
Again, the answer is: 1.17359909646538, -2.84026576313205. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B5%2Ax%2B-10+%29

Therefore, x = {1.1736,-2.8403}

Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
log x + log (3x + 5)= 1
log x(3x + 5)= 1
x(3x + 5)= 10^1
x(3x + 5)= 10
3x^2 + 5x= 10
3x^2 + 5x - 10 = 0
Applying the "quadratic formula" yields:
x = {1.17, -2.84}
throw out the negative solution (extraneous) leaving:
x = 1.17
.
Details of "quadratic formula" follows:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B5x%2B-10+=+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=%285%29%5E2-4%2A3%2A-10=145.

Discriminant d=145 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-5%2B-sqrt%28+145+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%285%29%2Bsqrt%28+145+%29%29%2F2%5C3+=+1.17359909646538
x%5B2%5D+=+%28-%285%29-sqrt%28+145+%29%29%2F2%5C3+=+-2.84026576313205

Quadratic expression 3x%5E2%2B5x%2B-10 can be factored:
3x%5E2%2B5x%2B-10+=+3%28x-1.17359909646538%29%2A%28x--2.84026576313205%29
Again, the answer is: 1.17359909646538, -2.84026576313205. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B5%2Ax%2B-10+%29