How to solve $x^x=100$?

Question

$x^x = 100$.

I have no clue on how to solve this. If you guys have, please show me your solution as well.

The solution is about $3.6$. I'm certain that it cannot be solved in closed form, without the use of a special function. — Mark McClure, Jan 01 '15 at 09:08
What kind of a solution are you looking for? Would it be enough to prove that there is only one positive real solution and calculate an approximate value for it? Where does $x$ take values? — Joonas Ilmavirta, Jan 01 '15 at 09:09
Have a look at the following. http://math.stackexchange.com/questions/50316/xx-y-how-to-solve-for-x — Radz, Jan 01 '15 at 09:10
There is closed form; to solve it using graphics, take logarithms and plot $x\log(x)-\log(100)$. — Claude Leibovici, Jan 01 '15 at 09:15

Teoc · Answer 1 · 2015-01-01T09:23:58.863

11

$$\begin{align} x^x&=100\\ x\log x&=\log 100\\ e^{\log x}\log x&=\log 100\\ \log x&=W(\log 100)\\ x&=e^{W(\log 100)}\\ x&\approx 3.597 \end{align}$$ Where $W(x)$ is the ProductLog function, defined as the inverse of $[f(x)=xe^x]$

Explanations: 1. Problem

2.Logs

3.$ e^{\log x}=x$

4.Definition of $W(x)$

5.Taking $e^x$

6.Numerical Solution

edited Jan 01 '15 at 09:23

answered Jan 01 '15 at 09:14

Teoc

8,700

He explains in the answer. Read the last line? – H_B Jan 01 '15 at 09:17
Don't you mean $e^{\ln(x)} = x$? You wrote log instead of ln. – DrZ214 Jan 31 '17 at 09:35
2

@DrZ214 $\log$ is commonly used to denote the base e logarithm. You really only see log denoting the base 10 log in high school. – Teoc Jan 31 '17 at 15:31
@Displayname I don't think I've ever seen it used that way before. I live in the states and got through calc 1/2/3 in undergrad, if it matters. – DrZ214 Jan 31 '17 at 22:45

score 4 · Answer 2 · answered Jan 01 '15 at 09:20

$$3^3=27\,\,\,\,\,\,\,\,\,\,\,\,\,\ 4^4=256$$ Therefore there are no integer solutions for this equation. The unique solution of this equation with $50$ decimal places can obtain using Mathematica as

$$3.5972850235404175054976522517822860691355430548866.$$

Also I plot the graph of $f(x)=x^x-100$ in $x\in[3,4].$

enter image description here

Adam Boit · Answer 3 · 2019-02-02T10:12:32.113

1

x=log(100)/log(x)

x'=x/2+log(100)/2log(x)

After several iteration for x (that is x', x'' to xn) you'll come to a good value of x to so much decimal places.

What am doing up there is applying the Babylonian method (an iterative algorithm similar to Newton-Raphson method). I've reduced it to its empirical form. Self taught I should add.

edited Feb 02 '19 at 10:12

answered Feb 02 '19 at 04:54

Adam Boit

21

1

Please format your answer for better clarity. Also, please explain that you're proposing a numerical approximation via an iterative method. Thanks! – Stefan Lafon Feb 02 '19 at 05:17
This is a good answer that contributes to the question what other answers have not! You can make it even better by formatting your answer, and using mathjax to type your math expressions to improve readability. Cheers! – YiFan Tey Feb 02 '19 at 05:48
I've no access to a computer (I'm using a feature phone) so formatting is not possible unfortunately. – Adam Boit Feb 02 '19 at 09:07

How to solve $x^x=100$?

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