# 2^x = x

1. Apr 3, 2004

### Russell E. Rierson

If 2^x = x

then

2^(2^x) = x

also!

2^2^(2^x) = x

2^2^2^(2^x) = x

2^2^2^2^2^2^2^ ...^(2^x)_n = x

2. Apr 3, 2004

### NateTG

Hmm.
$$\frac{d}{dx} 2^x = \ln 2 2^x$$
which is monotone increasing and
$$\frac{d}{dx} x = 1$$
which is constant, so if
$$2^x > x$$
where
$$\ln 2 \times 2^x = 1 \rightarrow x= -\log_2 ({\ln 2}) \approx 0.5$$
then the only solutions are imaginary.
So I don't think there are any real solutions.

3. Apr 3, 2004

### philosophking

Huh?

If 2^x=x, then

2^(2^x) does not equal x, it equals 2^x; you have to do the same thing to both sides, right??? I guess I don't understand your question then. It definitely does not have real solutions, because of the reason above.

4. Apr 3, 2004

### Hurkyl

Staff Emeritus
Sure, 2^(2^x) equals 2^x... but what does 2^x equal?

5. Apr 3, 2004

### philosophking

Ah i see, defined recursively

6. Apr 3, 2004

### phoenixthoth

x=-LambertW(-log2)/log2, i think...

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