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solving in x involving both exponential and logarithmic function

Is it possible to solve a function with both exponential and logarithm such as $a x^2 - b.\log(x) = c$ in closed form; where $a,b,c$ are constants and $a>0$ and $b>0$?
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When is the degree of this number 3?

I am helping a friend of mine, that works in history of mathematics. She is studying the story of the solution of the cubic equation by Cardano. Sometimes she asks me some mathematical questions, that ...
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An equation about generating functions and subfactorial

As I promised, I clone the problem from Math.SE to here, in order to find a solution. Suppose $G_n(w)$ is a formal power series (really a probability generating function, see the following ...
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expanding the sqare of sum

If there any way to expand the following? $$\left(\sum_{i=1}^nx_i\right)^{\frac{1}{2}}$$ and more generally, a way to expand $$\left(\sum_{i=1}^nx_i\right)^{\frac{p}{q}}$$ where $gcd(p,q) = 1$ ...
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Can roots of any polynomial be expressed using Eulerian function?

I encountered an interesting function which is called "Eulerian" by the Wolfram's MathWorld: $$\phi(q)=\prod_{k=1}^{\infty} (1-q^{k})$$ It is interesting because it is claimed that roots of any ...
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For any $n$, does there exist a number field with at least $n$ solutions to the unit equation

Let $n$ be a positive integer. Does there exist a number field $K$ such that the number of solutions of the unit equation $$a+b =1, \quad a,b\in O_{K}^\ast$$ is at least $n$? Can we write down such a ...
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satisfiable polynomial equations for given free coefficients

Let $F$ be a finite field, $n, k, m$ be natural numbers. I give you $m$ vectors $c^{(1)},\ldots,c^{(m)}\in F^n$. I ask for polynomials $p_1,\ldots,p_n$ on $k$ variables over $F$ such that the system ...
I recently came across some polynomials with some remarkable properties. A polynomial $P(u,v) \in \mathbb{R}[u,v]$ in 2 variables is remarkable if the set of solutions to the system ...