SOLUTION: what is log7 x + log7 (4x+21) = 3?

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Question 195221: what is log7 x + log7 (4x+21) = 3?
Found 2 solutions by solver91311, nerdybill:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


It is a logarithmic equation.


John

Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
Applying "log rules":
log7 x + log7 (4x+21) = 3
log7 x(4x+21) = 3
x(4x+21) = 7^3
4x^2 + 84 = 343
4x^2 + 84 - 343 = 0
Solving using the quadratic equation yields:
x = {3.5, -24.5}
Throw out the negative solution leaving:
x = 3.5
.
Details of quadratic follows:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 4x%5E2%2B84x%2B-343+=+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=%2884%29%5E2-4%2A4%2A-343=12544.

Discriminant d=12544 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-84%2B-sqrt%28+12544+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2884%29%2Bsqrt%28+12544+%29%29%2F2%5C4+=+3.5
x%5B2%5D+=+%28-%2884%29-sqrt%28+12544+%29%29%2F2%5C4+=+-24.5

Quadratic expression 4x%5E2%2B84x%2B-343 can be factored:
4x%5E2%2B84x%2B-343+=+4%28x-3.5%29%2A%28x--24.5%29
Again, the answer is: 3.5, -24.5. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B84%2Ax%2B-343+%29