SOLUTION: {{{ 2log(9,x) = 1/2 + log(9,(5x + 18)) }}}

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Question 691401: +2log%289%2Cx%29+=+1%2F2+%2B+log%289%2C%285x+%2B+18%29%29+
Found 2 solutions by nerdybill, mouk:
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
2log(base9)x = 1/2 + log(base9)(5x + 18)
log(base9)x^2 = 1/2 + log(base9)(5x + 18)
log(base9)x^2 - log(base9)(5x + 18) = 1/2
log(base9)x^2/(5x + 18) = 1/2
x^2/(5x + 18) = 9^(1/2)
x^2/(5x + 18) = 3
x^2 = 3(5x + 18)
x^2 = 15x + 46
x^2-15X = 46
x^2-15X-46 = 0
Applying the "quadratic formula" yields:
x = {17.61, -2.61}
throw out the negative solution (extraneous) leaving:
x = 17.61
.
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%2B-15x%2B-46+=+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=%28-15%29%5E2-4%2A1%2A-46=409.

Discriminant d=409 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--15%2B-sqrt%28+409+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-15%29%2Bsqrt%28+409+%29%29%2F2%5C1+=+17.6118742080783
x%5B2%5D+=+%28-%28-15%29-sqrt%28+409+%29%29%2F2%5C1+=+-2.61187420807834

Quadratic expression 1x%5E2%2B-15x%2B-46 can be factored:
1x%5E2%2B-15x%2B-46+=+1%28x-17.6118742080783%29%2A%28x--2.61187420807834%29
Again, the answer is: 17.6118742080783, -2.61187420807834. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-15%2Ax%2B-46+%29

Answer by mouk(232) About Me  (Show Source):
You can put this solution on YOUR website!
+2log%289%2Cx%29+=+1%2F2+%2B+log%289%2C%285x+%2B+18%29%29+
+2log%289%2Cx%29+-+log%289%2C%285x+%2B+18%29%29+=+1%2F2+
+log%289%2Cx%5E2%29+-+log%289%2C%285x+%2B+18%29%29+=+1%2F2+ (using +alog%28b%29=log%28b%5Ea%29+)
+log%289%2Cx%5E2+%2F+%285x+%2B+18%29%29+=+1%2F2+ (using +log%28a%29-log%28b%29=log%28a%2Fb%29+)
+x%5E2+%2F+%285x+%2B+18%29+=+9%5E%281%2F2%29+ (taking anti-logarithms)
+x%5E2+%2F+%285x+%2B+18%29+=+sqrt%289%29+ (definition of fractional powers)
+x%5E2+%2F+%285x+%2B+18%29+=+3+
+x%5E2+=+3%285x+%2B+18%29+
+x%5E2+=+15x+%2B+54+
+x%5E2+-+15x+-+54+=+0+ (which is a quadratic in x ...)
+%28x-18%29%28x%2B3%29+=+0+ (... and it factorises very nicely)
+x=-3+ or +x=18+
Now original question involved +2log%289%2Cx%29+ and you cannot take the logarithm of a negative number, so we can assume that +x+%3E+0+ and eliminate any negative solutions.
So, answer is +x=18+