Intereting Posts

Closed form for $\sum_{n=0}^\infty\frac{\operatorname{Li}_{1/2}\left(-2^{-2^{-n}}\right)}{\sqrt{2^n}}$
Property of Derivative in a local field
Prove the inequality for composite numbers
Flows of $f$-related vector fields
Rolling ellipses
Finding an asymptotic for the sum $\sum_{p\leq x}p^m$
Any example of manifold without global trivialization of tangent bundle
Finding closed forms for $\sum n z^{n}$ and $\sum n^{2} z^{n}$
Sums of complex numbers – proof in Rudin's book
Least Impossible Subset Sum
Show that if $r$ is an nth root of $1$ and $r\ne1$, then $1 + r + r^2 + … + r^{n-1} = 0$.
When is a function satisfying the Cauchy-Riemann equations holomorphic?
Lie group structure on some topological spaces
Is there a proof that $\pi \times e$ is irrational?
If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f (x + y) = f (x) f (y)$ and continuous at $0$, then continuous everywhere

How to solve this equation $$x^{2}=2^{x}$$

where $x \in \mathbb{R}$.

Por tentativa erro consegui descobri que $2$ é uma solução, mas não encontrei um método pra isso. Alguma sugestão?(*)

- Prove that the limit definition of the exponential function implies its infinite series definition.
- Is the natural logarithm actually unique as a multiplier?
- Reason for LCM of all numbers from 1 .. n equals roughly $e^n$
- Rational Exponent
- For which complex $a,\,b,\,c$ does $(a^b)^c=a^{bc}$ hold?
- Proving the irrationality of $e^n$

(Translation: By trying different values I’ve found that $2$ is a solution, but I couldn’t find any method to this though. Any suggestions? )

- Integration by parts: $\int e^{ax}\cos(bx)\,dx$
- Intuitive Understanding of the constant “$e$”
- For which $x$ is $e^x$ rational? Transcendental?
- Is showing $\lim_{z \to \infty} (1+\frac{1}{z})^z$ exists the same as $\lim_{n \to \infty} (1+1/n)^n$ exists
- If $a+b=2$ so $a^a+b^b+3\sqrt{a^2b^2}\geq5$
- How to understand why $x^0 = 1$, where $x$ is any real number?
- Solving equation involving self-exponentiation
- Since $2^n = O(2^{n-1})$, does the transitivity of $O$ imply $2^n=O(1)$?
- How did Bernoulli approximate $e$?
- Where we have used the condition that $ST=TS$, i.e, commutativity?

The equation can be written $x\log2=2\log|x|$. Let’s consider the function

$$

f(x)=x\log2-2\log|x|

$$

defined for $x\ne0$.

We have easily

$$

\lim_{x\to-\infty}f(x)=-\infty,

\qquad

\lim_{x\to\infty}f(x)=\infty

$$

and

$$

\lim_{x\to0}f(x)=\infty.

$$

Moreover

$$

f'(x)=\log2-\frac{2}{x}=\frac{x\log2-2}{x}

$$

Set $\alpha=2/\log2$; then $f'(x)$ is positive for $x<0$ and for $x>\alpha$, while it’s negative for $0<x<\alpha$.

Thus the function is increasing in $(-\infty,0)$, which accounts for a solution in this interval. In the interval $(0,\infty)$ the function has a minimum at $\alpha$ and

$$

f(\alpha)=\frac{2}{\log2}\log2-2\log\frac{2}{\log2}

=2(1-\log2+\log\log2)\approx-0.85

$$

Since the minimum is negative, this accounts for two solutions in $(0,\infty)$, which clearly are $x=2$ and $x=4$.

Try this, first suppose $x > 0$, then you take $x<0$.

$$\begin{align}x^2 = 2^x &\Rightarrow (x^2)^{\frac{1}{2}} = (2^x)^\frac{1}{2} \Rightarrow x= 2^\frac{x}{2} \\ & \Rightarrow x \ e^{-x\frac{\ln\ 2}{2}} = 1 \Rightarrow -x \frac{\ln\ 2}{2}\ e^{-x\frac{\ln\ 2}{2}} = -\frac{\ln \ 2}{2} \\ &\Rightarrow -x \frac{\ln\ 2}{2} = W(-\frac{\ln\ 2}{2}) \Rightarrow x = -\frac{2\ W(-\frac{\ln \ 2}{2})}{\ln\ 2}\end{align}$$

Which gives us

$x = -\frac{2\ W(\frac{-\ln \ 2}{2})}{\ln\ 2} = 2$ , in case $x > 0$

Similarly we may find

$x = -\frac{2\ W(\frac{\ln \ 2}{2})}{\ln\ 2} \approx -0,76666$, in case $x < 0$

Where $W$ is the Lambert’s funtion.

I wrote about this in another post.

The other solution is 4.

For $x^a = a^x; a > 0$ in general… Well, notice the shapes of the graphs for $x \ge 0$ are such that they intersect twice if $a \ne e$ and intersect once if $a = 1$ (at $x = e$).

In my other post I went into great detail about taking the first and second derivatives to show both $a^x$ and $x^a$ are concave and that they can only have at most two intersections and as $a^0 = 1 > 0^a = 0$ there must have at least one. But I’ll leave that as an excercise to the reader this time.

But obviously $x = a \implies a^a = a^a$ is a solution. The other … hmm… I forget and I’m too lazy to figure it out a second time. But the solutions are “paired” (that is if $x = b$ solves $a^x = x^a$ then $x = a$ solves $b^x = x^b$) and one solution is less than e and the other is greater than e. I came up with a formula relating the solutions.

2 and 4 solves both $x^2 = 2^x$ and $x^4 = 4^x$ were the slickest solutions but every other positive value for $a$ and a $a$ and $a’$ solution.

If $a$ is even or an even rational then there is an $x = -1/a$ solution. But if a is odd then there is no negative solution as for $x < 0$ then $a^x > 0$ but $x^a < 0$. If a is irrational then $x^a$ is undefined for $x < 0$. These are the only solutions as $a^x$ is increasing but $x^a$ is decreasing when $x < 0$ and $a$ is even.

- $\mathbb Z^n/\langle (a,…,a) \rangle \cong \mathbb Z^{n-1} \oplus \mathbb Z/\langle a \rangle$
- What is the relationship between $(u\times v)\times w$ and $u\times(v\times w)$?
- Examples of measurable and non measurable functions
- Origins of the modern definition of topology
- Counting square free numbers co-prime to $m$
- find the measure of $AMC$
- What do you call numbers such as $100, 200, 500, 1000, 10000, 50000$ as opposed to $370, 14, 4500, 59000$
- Finding a common factor of two coprime polynomials
- Minimum multi-subset sum to a target
- $\int \frac{\sin^3x}{\sin^3x + \cos^3x)}$?
- For any integer n greater than 1, $4^n+n^4$ is never a prime number.
- show that if $2^n -1$ is prime than n is also prime
- Finding the sum of $\sin(0^\circ) + \sin(1^\circ) + \sin(2^\circ) + \cdots +\sin(180^\circ)$
- Sequence problem involving inequalities
- Help on solving an apparently simple differential equation