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 $\frac{1}{y} \dydx = \ln(x) + 1$, so $\dydx = x^x \br{\ln(x)+1}$.

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