SOLUTION: Solve the logarithmic equation for x. (Round your answer to four decimal places.) ln(x − 6) + ln(x + 7) = 1

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Solve the logarithmic equation for x. (Round your answer to four decimal places.) ln(x − 6) + ln(x + 7) = 1       Log On


   

Question 446447: Solve the logarithmic equation for x. (Round your answer to four decimal places.)
ln(x − 6) + ln(x + 7) = 1
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
ln(x − 6) + ln(x + 7) = 1
ln(x − 6)(x + 7) = 1
(x − 6)(x + 7) = e^1
x^2+x-42 = e^1
x^2+x-42-e^1 = 0
x^2+x-42-2.718282 = 0
x^2+x-44.718282 = 0
apply the quadratic formula to get:
x = {6.2058, -7.2058}
the negative solution is extraneous -- throw it out leaving:
x = 6.2058
.
Details of quadratic follows:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B1x%2B-44.71828183+=+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-44.71828183=179.87312732.

Discriminant d=179.87312732 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-1%2B-sqrt%28+179.87312732+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%281%29%2Bsqrt%28+179.87312732+%29%29%2F2%5C1+=+6.20583938295572
x%5B2%5D+=+%28-%281%29-sqrt%28+179.87312732+%29%29%2F2%5C1+=+-7.20583938295572

Quadratic expression 1x%5E2%2B1x%2B-44.71828183 can be factored:
1x%5E2%2B1x%2B-44.71828183+=+1%28x-6.20583938295572%29%2A%28x--7.20583938295572%29
Again, the answer is: 6.20583938295572, -7.20583938295572. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B1%2Ax%2B-44.71828183+%29