janmr blog

Which quantity is larger, $a^b$ or $b^a$, for positive numbers $a$ and $b$? As it turns out, sometimes $a^b$ is the larger number, other times it is $b^a$, and it may even be the case that the two quantities are equal, even though $a \neq b$.

Let us start by rewriting the inequality:

$$\begin{aligned} a^b &< b^a \quad &\Leftrightarrow \\ e^{b \ln a} &< e^{a \ln b} \quad &\Leftrightarrow \\ b \ln a &< a \ln b \quad &\Leftrightarrow \\[6pt] \frac{\ln a}{a} &< \frac{\ln b}{b} \; . \end{aligned}$$

(Here we use the fact that $x \mapsto e^x$ is a (strictly) increasing function and that $a$ and $b$ are positive.)

The variables have now been separated and let us introduce $s(x) = \frac{\ln x}{x}$ for $x > 0$. Note how $s(a) < s(b) \Leftrightarrow a^b < b^a$ for all $a, b > 0$ (and similarly for the relations $=$ and $>$).

We see that $s$ is continuous on the positive real numbers, that $s(1)=0$, that $s(x)<0$ for $0<x<1$ and that $s(x)>0$ for $x>1$. This gives us our first, partial, result:

$$0 < a < 1 \leq b \quad \Leftrightarrow \quad s(a) < s(b) \quad \Leftrightarrow \quad a^b < b^a \; .$$

The function $s$ is, in fact, differentiable on the positive reals and the derivative can give us all the information we need:

$$s'(x) = \frac{\frac{1}{x} \cdot x - \ln x \cdot 1}{x^2} = \frac{1 - \ln x}{x^2} \; .$$

We see that $s'(x)>0$ for $0 < x < e$, $s'(e)=0$, and $s'(x)<0$ for $x > e$. See a plot of $x \mapsto s(x)$ in Figure 1.

From the information of the derivative of $s$, we have the following:

$$\begin{aligned} 0 < a < b \leq e \quad &\Leftrightarrow \quad s(a) < s(b) \quad &\Leftrightarrow \quad a^b < b^a \; , \\ e \leq a < b \quad &\Leftrightarrow \quad s(a) > s(b) \quad &\Leftrightarrow \quad a^b > b^a \; . \end{aligned}$$

To summarize, we now have the following relations, provided that $a < b$:

The only unresolved case is when $1 \leq a < e < b$. In fact, given the fact that $s(x) \to 0$ as $x \to \infty$, we have that $s(a) = s(b)$ has infinitely many solutions with $1 < a < e$ and $b > e$. Or, equivalently:

$$a^b = b^a \text{ has infinitely many solutions for $1 < a < e$ and $b > e$.}$$

Let us conclude by considering integer $a$ and $b$.

The case $a=1$ is trivial ($1^b < b^1$ for all $b > 1$).

The case $a=2$ is especially interesting because 2 is the only integer between 1 and $e$. This means, as argued above, that there is a nontrivial solution to $a^b=b^a$ and, in fact, $2^4=4^2$. So

$$\begin{aligned} 2^3 &< 3^2, \\ 2^4 &= 4^2, \\ 2^b &> b^2 \quad \text{for all integer $b \geq 3$.} \end{aligned}$$

When $a \geq 3$ it is more simple as we are in the $e < a < b$ category. We have

$$a^b > b^a \quad \text{for integer $a \geq 3$ and integer $b > a$.}$$