Finding solution in naturals to $a^b=b^a$ with the help of $f(x)=\frac{\ln(x)}{x}$

As the title said,

I'm trying to find solution in naturals numbers to $a^b=b^a$ with the help of the function

$f(x)=\large\frac{\ln(x)}{x}$.

I've been reading some solutions of that problem posted in math forums, and still don't know how to deduce, the obvious solutions, $(2,4)$ and $(4,2).$

Thank you very much!

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The missing ingredient in the answers so far is this interpretation: carefully fraw the curve $y = \log x.$ Your $f(a)$ is the slope of the line between the origin $(0,0)$ and the point on the graph $(a, \log a).$ Then, if $f(a) = f(b)$ for any real numbers $a,b,$ this means the slopes are the same, hence there is a straight line going through the points $(0,0),(a, \log a),(b, \log b).$

If you look a little more carefully, you see that the line through the origin with slope $1/e$ is tangent to the curve at $(e,1),$ which uses calculus. The slope (positive) of any line that intersects the curve twice is $0 < m < 1/e.$ Finally, given the two intersections $(a, \log a),(b, \log b)$ with $a < b,$ we find that $$1 < a < e.$$ Since you wanted $a$ to be an integer, it follows that $a=2.$

Please draw the graph and some lines through the origin. It becomes quite easy to see, with a careful graph, that the smaller one (real number $a$) satisfies $1 < a < e.$ I ran out of energy. In the image below, the graph in black is $y = \log x,$ the green line is the tangent to the graph that goes through the origin and the point $(e,1),$ and the blue line goes through the origin and $(2, \log 2)$ and $(4,\log 4).$ If you think about it, $\log 4$ is exactly double $\log 2,$ so we are looking at some similar triangles. I do not remember whether there are any other rational solutions to this.

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+1 Very nicely put! – amWhy Mar 9 '13 at 0:33

Knowing that when $x > 0,\;\; x^y = e^{\large \ln x^y}= e^{y \ln x}$,

we have, since $a, b \in \mathbb{N}:$ $$a^b = b^a \iff e^{b\ln a} = e^{a \ln b} \iff b \ln a = a\ln b$$

Now try dividing each side of the equation by $\;ab$.

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After dividing: $f(a)=f(b)$, and now what? hmm... – 17SI.34SA Mar 8 '13 at 21:02
@amWhy, could you please elaborate more? Clearly I'm missing were something :( – 17SI.34SA Mar 8 '13 at 21:23
$f(a) = f(b) \implies a = b \;\;(a, b \geq 1),\;$ or $\;a = 2, b = 4\;$ or $\;a = 4, b = 2$ – amWhy Mar 8 '13 at 21:26
See Wolfram for a graph of $(\ln x)/x$ for x>1. You'll see that the only positive INTEGER values $a\neq b$ for which $f(a) = f(b)$ are $(2, 4), (4, 2).$ – amWhy Mar 8 '13 at 21:34

Note that $$f'(x)=\frac{\mathrm{d}}{\mathrm{d}x}\frac{\log(x)}{x}=\frac{1-\log(x)}{x^2}\tag{1}$$ Suppose that $a\lt b$ and $a^b=b^a$. Taking $\log$ and dividing by $ab$ yields $$\frac{\log(a)}{a}=\frac{\log(b)}{b}\tag{2}$$ $(1)$, $(2)$, and Rolle's Theorem say that there must be an $x$ between $a$ and $b$ so that $$f'(x)=\frac{1-\log(x)}{x^2}=0\tag{3}$$ $(3)$ is satisfied if and only if $x=e$. Therefore, if $a\lt b$, then $$a^b=b^a\Rightarrow a\lt e\lt b\tag{4}$$ Thus, there are two options for $a\in\mathbb{Z}$, $1$ and $2$. Since $1^b=b^1\Rightarrow b=1$, the only solution with $a<b$ must have $a=2$. Since $$\frac{\log(2)}{2}=\frac{2\log(2)}{2\cdot2}=\frac{\log(4)}{4}\tag{5}$$ we must have $b=4$. Any other solution, $x$, of $\frac{\log(2)}{2}=\frac{\log(4)}{4}=\frac{\log(x)}{x}$ must have $e$ between $2$ and $x$ and $e$ between $x$ and $4$. Both cannot be true simultaneously for any $x$.

The only solution is therefore $$2^4=4^2\tag{6}$$

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We write the factor prime decompsition of $a$ and $b$ : $$a=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_r^{\alpha_r}\quad\text{and}\quad b=p_1^{\beta_1}p_2^{\beta_2}\cdots p_r^{\beta_r},$$ with $0\leq \alpha_i$ and $0\leq \beta_i$. The relation $a^b=b^a$ gives: $$\forall i\in\{1,2,\ldots,r\}, \alpha_ib=\beta_ia.$$ If we suppose $a\leq b$, then we have: $$\forall i\in\{1,2,\ldots,r\}, \alpha_ib\leq\beta_ia$$ so $a|b$.

Let $m$ defined by $b=ma$, then $a$ solve the equation: $$a^{m-1}=m.$$ Let's search the solutions of this equation.

We note that if $a\geq 2$ and $m>2$, then $a^{m-1}\geq 2^{m-1}>m$.

If $a\geq2$ is solution then $m\leq 2$, we find $a=2$ and $m=2$, $a\geq2$ and $m=1$ as solutions.

Finally the solutions are $(a,b)=(2,4)$ or $(a,b)=(4,2)$ or $a=b$.

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