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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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up vote 0 down vote accepted

First it should be said that $x^2$ is not exponential, since $x$ is your independent variable. An exponential terms generally means something of the form $e^{kx}$. Your equation is transcendental and often does not have a solution in algebraic numbers (unlike quadratic eqations, for instance).

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An exponential function is a function for which the variable is an exponent. For example $x \mapsto 2^x$ is an exponential function, as is $x \mapsto 7^{2x^3+4}$. Of course $2^x \equiv \operatorname{e}^{kx}$ when $k = \ln 2$. – Fly by Night Apr 14 '13 at 16:59

It depends what you mean by "solve". In general, not using elementary functions, no. The general solution to the problem you posted above involves Lambert's $W$-function.

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