Problem 10 fall 2011

Problem 9 is a little hard for me to solve, because I did not touch the series problem for at least 4 years. So I do problem 10 first.

There are lots of examples can be list here like this one. Let $a=\sqrt{3}, b=\log_{\sqrt{3}}2$, then
$$a^b=\sqrt{3} ^\log_{\sqrt{3}}2$$ $$ =2 $$
And it is easy know that $\sqrt{3}$ is a irrational number. The following to prove $\log_{\sqrt{3}}2$ is also a irrational number.
If it is not, then $\log_{\sqrt{3}}2 = \frac{p}{q}$, where $(p,q)=1, p,q \in \mathbb{N}$.
Therefore $\sqrt{3}^{p/q}=2$, which can be write as $\sqrt{3}^p=2^q$. Square both side, we get $3^q=4^p$.  Since $p,q$ are both nature number ($>0$), but R.H.S. is an odd number, L.H.S is an even number. Then the assumtion faile. Then $\log_{\sqrt{3}}2$ is an irrational number.