Significance of $1.32109977$ in the area of exponential/logarithmic functions I noticed that if you graph $y = log_n(x)$ and $y = x^n$ simultaneously for various values of $n$. Specifically, I noticed that with higher values of $n$, the graphs crossed twice. For lower values of $n$ (close to $1$) the graphs did not cross. I wondered at what point there would be exactly one shared point. I figured this out by using Wolfram Alpha. I inputted log base $n(x) = x^n$ where $n=<number>$, and tried to make the outputted solutions as close to each other as I can. I ended up with the number $1.32109977$. I'm curious as to 3 things:
1) What is the exact value of this number (if that can be found)?
2) Is this number at all relevant or just a value where a graph happens to look a certain way?
3) What is this number called? (...because out of a world of >7,000,000,000 people, someone has named this number)

According to alpha, a solution to $$\log_n(x)=x^n$$ is $$\sqrt[n]{-\frac{W(-n\log(n))}{n\log(n)}}$$
Here $W$ denotes the Lambert W function. You can also read about it on alpha. The argument of the $n$'th root is actually positive, provided $-n\log(n)\ge -\frac{1}{e}$ (apart from a removable singularity at $n=1$).  If the input to $W$ is less than $-1/e$, the result is non-real, and hence the solution will become complex.  Hence the critical value you seek is $$n\log n=1/e$$
This has exact value $$e^{W(1/e)}\approx 1.321099762015617456962355870878829561624$$
As far as I know this constant has no particular significance.
