SOLUTION: find x, Log{base x}5=8+Log{base 9}x for the left x is the base. for right 9 is the base.

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Question 449498: find x,
Log{base x}5=8+Log{base 9}x
for the left x is the base. for right 9 is the base.
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
log%28x%2C5%29+=+8+%2B+log%289%2Cx%29
log(5)/log(x) = 8 + log(x)/log(9) = [8*log(9) + log(x)]/log(9)
log(5)/log(x) = (8*log(9) + log(x))/log(9)
log(5)*log(9) = log(x)*(8log(9) + log(x))
%28log%28x%29%29%5E2+%2B+8%2Alog%289%29%2Alog%28x%29+-+log%285%29log%289%29+=+0
Sub u for log(x)
u%5E2+%2B+8%2Alog%289%29%2Au+-+log%285%29%2Alog%289%29+=+0
u%5E2+%2B+7.63394u+-+0.6669869+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B76.3394x%2B0.6669869+=+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=%2876.3394%29%5E2-4%2A1%2A0.6669869=5825.03604476.

Discriminant d=5825.03604476 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-76.3394%2B-sqrt%28+5825.03604476+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2876.3394%29%2Bsqrt%28+5825.03604476+%29%29%2F2%5C1+=+-0.00873812546124952
x%5B2%5D+=+%28-%2876.3394%29-sqrt%28+5825.03604476+%29%29%2F2%5C1+=+-76.3306618745387

Quadratic expression 1x%5E2%2B76.3394x%2B0.6669869 can be factored:
1x%5E2%2B76.3394x%2B0.6669869+=+%28x--0.00873812546124952%29%2A%28x--76.3306618745387%29
Again, the answer is: -0.00873812546124952, -76.3306618745387. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B76.3394%2Ax%2B0.6669869+%29

x1 and x2 are the logs of x.
x = 0.980080785
x = 4.670228502E-77