Ask Uncle Colin: A surprising order

Dear Uncle Colin,

How come ${0.3}^{0.3}>{0.4}^{0.4}$?

- Puzzling Over It, Some Surprisingly Ordered Numbers

Hi, POISSON, and thank you for your message!

It is a bit surprising, isn’t it? You would expect $x^x$ to increase everywhere, at first glance.

Why it doesn’t

We can see that this isn’t the case if we differentiate $y=x^x$ - or rather, $\ln (y)=x\ln (x)$, which is much more tractable. We get $\frac{1}{y}\frac{dy}{dx}=\ln (x)+1$, so $\frac{dy}{dx}=x^x(\ln (x)+1)$.

That clearly gives a turning point when $\ln (x)=-1$, which it does when $x=e^{-1}$ - bang in between your two values of $x$.

Why in particular

To compare $y_1={0.3}^{0.3}$ and $y_2={0.4}^{0.4}$, I might start by comparing their tenth powers. Bear with me: unlike $x^x$, $f(x)=x^{10}$ is a strictly increasing function (for $x\ge 0$), so whichever number has the bigger 10th power is the bigger number.

So, $y_1^{10}={0.3}^3$ and $y_2^{10}={0.4}^4$. Dealing with those as fraction, $y_1^{10}=\frac{3^3}{{10}^3}$ and $y_2^{10}=\frac{4^4}{{10}^4}$.

The first of those is $\frac{27}{1,000}$, and the second is $\frac{256}{10,000}$, which is slightly smaller - the underlying reason is that $4^4$ is less than ten times greater than $3^3$.

Hope that helps!

- Uncle Colin