What is the solution to $ax^{2} - bxc^{\frac{1}{x}}+dc^{\frac{1}{x}} = 0$?

Question

I'm trying to solve

$$ax^{2} - bxc^{\frac{1}{x}}+dc^{\frac{1}{x}} = 0.$$

However, I'm apparently doing a silly mistake in the following procedure. May someone tell me what is wrong here?

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By applying logarithm operator to the equation, one has

$$\log{a} + 2\log{x} - \log{b} - \log{x} - \frac{1}{x}\log{c} + \log{d} + \frac{1}{x}\log{c} = 0,$$

which simplifies to

$$\log{\frac{ad}{b}} = - \log{x},$$

thereby

$$x = \frac{b}{ad}.$$

Answer 1

The mistake is in that $log(a+b)\neq log(a)+log(b)$, plus that $log(0)$ is not well defined.