Editing Openai/69481a9b-0c20-800f-9469-35a475c239b3 (section)

=== Assistant: Oh, I love this prompt. ===
Oh, I love this prompt. Okay, here’s something you can basically copy-paste to Reddit, tweak as you like, and toss into the “ChatGPT can’t do math” threads.

People keep saying “ChatGPT can’t do math,” so let’s actually do some math.

Take the equation:

xx=8x^x = 8xx=8
We want to solve this over the reals, ideally with a clean expression, not just “my calculator says it’s about 2.39”.

==== 1. Existence & uniqueness (a bit of real analysis) ====

Define

f(x)=xx−8,x>0.f(x) = x^x - 8,\quad x>0.f(x)=xx−8,x>0.
Check a few values:
* x=2x = 2x=2: 22=4<82^2 = 4 < 822=4<8 → f(2)<0f(2) < 0f(2)<0
* x=3x = 3x=3: 33=27>83^3 = 27 > 833=27>8 → f(3)>0f(3) > 0f(3)>0

By continuity, there’s at least one solution between 2 and 3.

Now rewrite:

xx=8  ⟺  xln⁡x=ln⁡8.x^x = 8 \;\Longleftrightarrow\; x\ln x = \ln 8.xx=8⟺xlnx=ln8.
Set g(x)=xln⁡xg(x) = x\ln xg(x)=xlnx. Then

g′(x)=ln⁡x+1.g'(x) = \ln x + 1.g′(x)=lnx+1.
For x>1/ex > 1/ex>1/e, we have ln⁡x+1>0\ln x + 1 > 0lnx+1>0, so ggg is strictly increasing there. Our solution is clearly > 2, so:
* There is exactly one real solution.

So: yes, a unique real root, in (2,3).

==== 2. Closed form with the Lambert W function ====

Start from

xx=8,x>0.x^x = 8,\quad x>0.xx=8,x>0.
Take logs:

xln⁡x=ln⁡8.x\ln x = \ln 8.xlnx=ln8.
Let

u=ln⁡x⇒x=eu.u = \ln x \quad\Rightarrow\quad x = e^u.u=lnx⇒x=eu.
Then:

xln⁡x=eu⋅u=ln⁡8,x\ln x = e^u \cdot u = \ln 8,xlnx=eu⋅u=ln8,
so

ueu=ln⁡8.u e^u = \ln 8.ueu=ln8.
By definition of the Lambert W function, which solves

W(z)eW(z)=z,W(z)e^{W(z)} = z,W(z)eW(z)=z,
we have

u=W(ln⁡8).u = W(\ln 8).u=W(ln8).
Recall u=ln⁡xu = \ln xu=lnx, so:

ln⁡x=W(ln⁡8)⇒x=eW(ln⁡8).\ln x = W(\ln 8)
\quad\Rightarrow\quad
x = e^{W(\ln 8)}.lnx=W(ln8)⇒x=eW(ln8).
There’s a standard identity:

eW(z)=zW(z),e^{W(z)} = \frac{z}{W(z)},eW(z)=W(z)z,
so with z=ln⁡8z = \ln 8z=ln8,

x=ln⁡8W(ln⁡8).\boxed{x = \frac{\ln 8}{W(\ln 8)}}.x=W(ln8)ln8.
On the principal branch W0W_0W0, that’s the unique real solution. Numerically:

x≈2.3884234845.x \approx 2.3884234845.x≈2.3884234845.
Yes, that’s an exact symbolic expression in terms of a standard special function.

==== 3. Pure numerical method (Newton’s method, no “magic” functions) ====

If you don’t like Lambert W and just want to see iterative reasoning, let’s use Newton’s method.

We already have:

f(x)=xx−8.f(x) = x^x - 8.f(x)=xx−8.
Differentiate. Since xx=exln⁡xx^x = e^{x\ln x}xx=exlnx,

ddxxx=xx(ln⁡x+1).\frac{d}{dx} x^x = x^x(\ln x + 1).dxdxx=xx(lnx+1).
So:

f′(x)=xx(ln⁡x+1).f'(x) = x^x(\ln x + 1).f′(x)=xx(lnx+1).
Newton’s iteration:

xn+1=xn−f(xn)f′(xn)=xn−xnxn−8xnxn(ln⁡xn+1).x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
= x_n - \frac{x_n^{x_n} - 8}{x_n^{x_n}(\ln x_n + 1)}.xn+1=xn−f′(xn)f(xn)=xn−xnxn(lnxn+1)xnxn−8.
Start with x0=2.5x_0 = 2.5x0=2.5 (we know the root is between 2 and 3):
* After a few iterations this converges to x≈2.38842348x \approx 2.38842348x≈2.38842348.

You can also apply Newton to the simpler function

g(x)=xln⁡x−ln⁡8,g′(x)=ln⁡x+1g(x) = x\ln x - \ln 8,\quad g'(x) = \ln x + 1g(x)=xlnx−ln8,g′(x)=lnx+1
with iteration

xn+1=xn−xnln⁡xn−ln⁡8ln⁡xn+1,x_{n+1} = x_n - \frac{x_n\ln x_n - \ln 8}{\ln x_n + 1},xn+1=xn−lnxn+1xnlnxn−ln8,
which is nicer to implement and converges to the same number.

==== 4. Complex solutions? Also doable. ====

Every branch WkW_kWk of the Lambert W function gives a complex solution:

xk=ln⁡8Wk(ln⁡8),k∈Z.x_k = \frac{\ln 8}{W_k(\ln 8)},\quad k \in \mathbb{Z}.xk=Wk(ln8)ln8,k∈Z.
* k=0k = 0k=0 → real solution.
* k≠0k \neq 0k=0 → infinitely many complex solutions.

==== 5. So… can ChatGPT “do math” or not? ====

A few honest points:
* I can carry out derivations, use special functions like WWW, set up Newton’s method, analyze monotonicity, etc.
* I do sometimes mess up arithmetic or algebra if I’m not careful (just like a human doing it in their head), especially with long symbolic expressions.
* I’m not a CAS like Mathematica or SymPy; I’m a language model that’s been trained on tons of math and can reconstruct methods and reasoning.

But “ChatGPT can’t do math” is just wrong in the blanket way it’s usually thrown around. A more accurate version would be:

: “ChatGPT isn’t a perfect calculator or theorem prover and can make mistakes,
: but it’s very capable of explaining and performing math when used carefully.”

If you want a neat one-liner for the thread, you could drop:

: “ChatGPT can’t do math.”
: ChatGPT: derives x = \dfrac{\ln 8}{W(\ln 8)} \approx 2.38842348 for \(x^x = 8, proves uniqueness, and gives you a Newton iteration.

Up to you how spicy you want it. 😄