SOLUTION: Solve each equation. Show all steps and work. Round your answers to four decimal places. a. 6^2x=250 b. log_3(x-3)+log_3(x-7)= 4

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Solve each equation. Show all steps and work. Round your answers to four decimal places. a. 6^2x=250 b. log_3(x-3)+log_3(x-7)= 4       Log On


   

Question 1138737: Solve each equation. Show all steps and work. Round your answers to four decimal places.
a. 6^2x=250
b. log_3(x-3)+log_3(x-7)= 4

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
a. 6^2x=250
a. 6%5E2x=250
36x = 250
x = 250/36
x =~ 6.9444
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b. log_3(x-3)+log_3(x-7)= 4
If you mean base 3:
log%283%2C%28x-3%29%29+%2B+log%283%2C%28x-7%29%29+=+4
log%283%2C%28%28x-3%29%2A%28x-7%29%29%29+=+4+=+log%283%2C3%5E4%29+=+log%283%2C81%29
(x-3)*(x-7) = 81
x%5E2+-+10x+%2B+21+=+81
x%5E2+-+10x+-+60+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-10x%2B-60+=+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=%28-10%29%5E2-4%2A1%2A-60=340.

Discriminant d=340 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--10%2B-sqrt%28+340+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-10%29%2Bsqrt%28+340+%29%29%2F2%5C1+=+14.2195444572929
x%5B2%5D+=+%28-%28-10%29-sqrt%28+340+%29%29%2F2%5C1+=+-4.21954445729289

Quadratic expression 1x%5E2%2B-10x%2B-60 can be factored:
1x%5E2%2B-10x%2B-60+=+%28x-14.2195444572929%29%2A%28x--4.21954445729289%29
Again, the answer is: 14.2195444572929, -4.21954445729289. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-10%2Ax%2B-60+%29

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Ignore the negative solution.
Logs of negative #s are not allowed.