Intersection of a power function with a line: how to compute?

Question

How to compute $x$ from

$$q x^p = 1 - x$$

where $x$ and $q$ are positive, while $p$ is a real number?

When $p > 0$: it's two monotonic functions, one increasing and one decreasing, and having opposite relations on the ends - single solution. When $p \le 0$: second derivative is positive - up to two solutions.

And, since this equation is not even algebraic, I'm almost sure that $x$ may only be found, at best, as a power series, but not expressed finitely through elementary functions. If so, then what is the series? (Sure binary search may be employed, but a more elegant method is preferable.)

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Alternatively, how to compute $x$ and $y$ from

$$a \log|x| + b \log|y| = c$$

$$x + y = 1$$

where $\log x, \log y$, $a, b$, and $c$ are real?

Sure I could expand the logarithms into the series and perform something weird with the coefficients, but isn't there an easier and cheaper (and well-known) way? Since the problem may be so easily formulated, someone probably already solved it, but I couldn't find anything related.

Answer 1

Simply put, there is no 'general closed-form solution' to a polynomial of degree $5$ or greater.

Since $p>5$ is more than possible, or even where $p$ is not a whole number, the general solution to this problem cannot be solved for.

Answer 2

Since both sides of the equation have a simple derivative, The Newton-Raphson method would give a "simple" recursive law for x approaching the solution. I think that is the most one can look for.