How to solve problems such as $x = \log_2{x}$

Question

How does one go about solving $x = \log_2{x}$? Is there a technique to solve these sorts of problems?

Use the method of Newton - Raphson to find a root of the equation $x-\log_2x=0$. See http://en.wikipedia.org/wiki/Newton%27s_method — Elias Costa, Dec 04 '12 at 12:27

score 4 · Answer 1 · answered Dec 04 '12 at 12:21

4

There are no elementary formulas for such equations. Using the Lambert W function, the solution is $x = e^{-W(-\log2)}$.

answered Dec 04 '12 at 12:21

lhf

216,483

score 2 · Answer 2 · answered Dec 04 '12 at 12:22

2

This can be "solved" using the Lambert W function. The example on the wikipedia page is fundamentally the same as your question, after rewriting as $2^x=x$.

answered Dec 04 '12 at 12:22

Daryl

5,598

score 1 · Answer 3 · edited Dec 04 '12 at 12:36

1

There are. Look at it as an intersection of two functions - $$f(x)=x$$ and $$g(x)=\log_2(x)=\ln(x)/\ln(2)$$ both functions are monotone increasing. Find their derivatives, and check if they can ever meet, and how many times. Separate the search to $(0,1)$ and $(1,\infty)$.

edited Dec 04 '12 at 12:36

E.O.

6,942

answered Dec 04 '12 at 12:10

mousomer

395
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How to solve problems such as $x = \log_2{x}$

3 Answers3