Solving equations of type x <mrow class="MJX-TeXAtom-ORD"> 1 <mrow clas

Bailee Short

Bailee Short

Answered question


Solving equations of type x 1 / n = log n x
First, I'm a new person on this site, so please correct me if I'm asking the question in a wrong way.
I thought I'm not a big fan of maths, but recently I've stumbled upon one interesting fact, which I'm trying to find an explanation for. I've noticed that graphs of functions y = x 1 / n and y = log n x , where n is given and equal for both functions, always have 2 intersection points. This means, equation x 1 / n = log n x must have 2 solutions, at least it's what I see from the graphs.
I've tried to solve this equation analytically for some given n, like 4, but my skills are very rusty, and I cannot come up with anything. So I'm here for help, and my question(-s) are:
are these 2 functions always have 2 intersection points?
if yes, why, if not, when not?
how to solve equations like x 1 / n = log n x analytically?

Answer & Explanation



Beginner2022-06-29Added 22 answers

x n = log n x x = t n   } t = n ln n ln t ; t = e u e u = n ln n u
( u ) e u = ln n n u = W ( ln n n ) x = t n = ( e u ) n = e n u
x = exp ( n W ( ln n n ) )
where W is the Lambert W function.


Beginner2022-06-30Added 4 answers

They always have two intersection points. Let
f ( x ) = x 1 / n log n x .
lim x 0 + f ( x ) = lim x + f ( x ) = + .
f ( x ) = 1 n x 1 / n 1 1 x log n .
It is easy to see that f has a single zero, which is necessarily a minimum (one can check that f has the appropriate signs at both sides of this point, which is
x m = n n ( log n ) n .
We have
f ( x m ) = n log n n log n ( log n log log n ) < 0 ,
so the minimum is achieved below the x-axis. This shows that f intersects the x-axis twice.
As for an analytic solution, I don't think that's possible.

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