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

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Question 691401:
Found 2 solutions by nerdybill, mouk:
Answer by nerdybill(7384)   (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 (in our case ) has the following solutons:



For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=409 is greater than zero. That means that there are two solutions: .




Quadratic expression can be factored:

Again, the answer is: 17.6118742080783, -2.61187420807834. Here's your graph:
Answer by mouk(232)   (Show Source): You can put this solution on YOUR website!


(using )
(using )
(taking anti-logarithms)
(definition of fractional powers)



(which is a quadratic in x ...)
(... and it factorises very nicely)
or
Now original question involved and you cannot take the logarithm of a negative number, so we can assume that and eliminate any negative solutions.
So, answer is

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