Solving for x: a*log(x)+b*log(x)^2+c*x = 0

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Homework Statement


Given the equation
a log_{10}(x)^{2}+b log_{10}(x) + c x + d = 0
solve for x

Homework Equations


I don't think the quadratic equation will work here. There are a lot of equations at these two pages:

http://en.wikipedia.org/wiki/Logarithm
http://en.wikipedia.org/wiki/Logarithmic_identity

The Attempt at a Solution



I'm not sure where to start. I'm not sure that it is possible to solve. If it is possible, and it hasn't been solved before, then it might take some serious hacking.
 
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I doubt there is an explicit solution for this!
 
Because of the x both within and outside the logarithm, you won't be able find a simple "algebraic" expression for for x. You might use numerical solution or the Lambert W function.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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