19
votes
1answer
304 views

The positive root of the transcendental equation $\ln x-\sqrt{x-1}+1=0$

I numerically solved the transcendental equation $$\ln x-\sqrt{x-1}+1=0$$ and obtained an approximate value of its positive real root $$x \approx 14.498719188878466465738532142574796767250306535...$$ ...
0
votes
2answers
143 views

Solutions of $ \frac{1- x^{\alpha}}{1- x^{\beta}} = \gamma$

Would it be possible to give pieces of information about $x \in \left]0,1\right[$ such that: $$ \frac{1- x^{\alpha}}{1- x^{\beta}} = \gamma$$ where $\alpha > 0$, $\beta > 0$ and $\gamma > ...
0
votes
1answer
105 views

Is there an closed form solution to the equation: $ae^a=1$ [duplicate]

Possible Duplicate: Inverse of $y=xe^x$ Is there a closed form solution to the equation: $a e^a$ = 1 ?