# A Famous Equation, x^2 = 2^x

A recent episode of the popular YouTube channel "blackpenredpen" is about solutions to the famous equation

`x^2 = 2^x`

You should be able to see two solutions yourself, `x = 2` and `x = 4`, because `2^2 = 2^2` and `4^2 = 2^4`.

This post is about a third solution.

### Contents

#### Two graphs

The figure shows the graphs of `x^2` and `2^x` and their intersections. The narrow silver area is the only region the right half plane where `x^2` is greater than `2^x`.

We can see there are three intersections -- the two we knew about at `x = 2` and `x = 4` and a third one at a negative value of `x`. Blackpen carefully explains how to characterize this negative solution.

black_pen_plot

#### The negative solution

I was pleased to learn that our Symbolic Math Toolbox can find all three solutions, provided we indicate that we are only interested in real-valued solutions. (The behavior of `x^2 = 2^x` for complex `x` is a topic for another day.)

syms x real z = solve(x^2 == 2^x)

z = 2 4 -(2*lambertw(0, log(2)/2))/log(2)

We want to know more about that third solution.

z = z(3)

z = -(2*lambertw(0, log(2)/2))/log(2)

The function `lambertw(x)` involved in this solution is an old friend, the LambertW function.

#### Logarithm

The blue curve in the following figure is a plot of `exp(x)`. It goes to zero for negative `x` and grows exponentially for positive `x`. Imagine interchanging the x and y axes by reflecting the blue curve about the diagonal dashed line to produce the orange curve. The orange curve is familiar. It is a plot of the functional inverse of `exp(x)` which we know as `log(x)`. If `y = log(x)` then `x = exp(y)`.

log_plot

Current plot released

#### LambertW

The blue curve in the following figure is now a plot of `x*exp(x)`. Reflecting the blue curve about the diagonal produces the orange curve, a plot of the functional inverse of `x*exp(x)`. This function is not as familiar as `log(x)`. It is `lambertw(x)`. If `y = lambertw(x)` then `x = y*exp(y)`.

We need to evaluate `y = lambertw(x)` at `x = log(2)/2`. This is the black dot. Once we have `x` and `y`, the negative solution to our famous equation is simply `z = -y/x`.

lambertw_plot

Current plot released

#### Many digits

The Symbolic Math Toolbox variable precision arithmetic, `vpa`, can produce the numeric value of the negative solution to any number of digits.

z vpaz = vpa(z,75)

z = -(2*lambertw(0, log(2)/2))/log(2) vpaz = -0.766664695962123093111204422510314848006675346669832058460884376935552795725

#### Fixed points

Here is your homework. Investigate the iteration:

`x = sign(x)*2^(x/2)`

Consider three situations:

- Starting values between 0 and 4.
- Starting values greater than 4.
- Starting values less than 0.

#### Thanks

Thanks to Mark Round for suggesting only real solutions.

#### Code

The code for the figures is available.

**Category:**- History,
- Numerical Analysis,
- People

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