How to solve infinite repeating exponents

exponentiation puzzle

How do you approach a problem like (solve for $x$):

$$x^{x^{x^{x^{...}}}}=2$$

Also, I have no idea what to tag this as.

Thanks for any help.


Solution 1:

I'm just going to give you a HUGE hint. and you'll get it right way. Let $f(x)$ be the left hand expression. Clearly, we have that the left hand side is equal to $x^{f(x)}$. Now, see what you can do with it.

Related

Trig Fresnel Integral
How to sum $\frac1{1\cdot 2\cdot 3\cdot 4} + \frac4{3\cdot 4\cdot 5\cdot 6} + \frac9{5\cdot 6\cdot 7\cdot 8} + \cdots$ quickly?
Decomposition of functionals on sobolev spaces
Calculate $\ln(2)$ using Riemann sum. [duplicate]
Complex derivative in terms of partial derivatives
$A,B\in M_{n}(\mathbb{R})$ so that $A>0, B>0$, prove that $\det (A+B)>\max (\det(A), \det(B))$
Differentiation under integral sign (Gamma function)
Proving that:$\int_X f_n g \, d\mu \to \int_X fg \, d\mu$ for all $g$ in $\mathscr{L}^q (X)$
Summation of $\sum\limits_{n=1}^{\infty} \frac{x(x+1) \cdots (x+n-1)}{y(y+1) \cdots (y+n-1)}$
Let $f : \mathbb{R} \to \mathbb{R}$ be measurable and let $Z = {\{x : f'(x)=0}\}$. Prove that $λ(f(Z)) = 0$.
Prove that if $R$ is an integral domain and has ACCP, then $R[X]$ has ACCP
limsup and liminf of a sequence of subsets of a set