TITLE: How to find out which number is larger without a calculator?
QUESTION [17 upvotes]: So I have a question which is:
Which is larger?
$$2.2^{3.3} \text{ or   } 3.3^{2.2} $$
Now I need to find out with using a calculator but the answer is $3.3^{2.2}$.
The only thing I could think of is rounding.
So you know:
$2^3=8$ and $3^2=9$
I'm interested in seeing if there are other ways just because there might be the possiblity of being asked:
Which is larger?
$$2.5^{3.5} \text{ or   } 3.5^{2.5} $$
So if I use the rounding idea, would I just round normally. I would get:
$$ 3^4 \text{ or } 4^3 $$
which shows $3^4$ is larger.
This is the only way I could think of, are there any other ways without using a calculator to determine which is larger?

REPLY [5 votes]: Another method is to use the fact that $\log a^b=b\log a$, so that you can reduce the exponents even if they don't have any obvious common divisors:
$$2.2^{3.3}\gtrless3.3^{2.2}$$
$$3.3 \log 2.2 \gtrless 2.2 \log 3.3$$
$$1.5 \log 2.2 \gtrless \log 3.3$$
$$2.2^{1.5} \gtrless 3.3$$
$$2.2 \cdot 2.2^{0.5} \gtrless 3.3$$
$$2.2^{0.5} \gtrless 1.5$$
$$(2.2^{0.5})^2 \gtrless (1.5)^2$$
$$2.2 < 2.25$$
$$\Rightarrow 2.2^{3.3}\lt3.3^{2.2}$$