Prove the next numbers are irrational

Question

I need to prove the next numbers are irrational:

$$ \sqrt{2}+\sqrt{5} $$ $$\sqrt{3}$$ $$ \log_{2}3 $$

Of course, I tried to prove it with negative evidence...

Answer 1

For first two, look at the links below (in comments). I am writing a short proof for the third.

Assume on the contrary it is rational, that is $$ \log_{2}3 =p/q $$ where $p$ and $q$ are integers and $q\neq 0$.

Now, multiplying $q$ on both sides, we get $$q \log_{2}3 =p$$

It implies, $2^{p}=3^{q}$.

We arrive at the contradiction as left hand side is even but right hand side is odd.