1. ## log proof

prove log base 2 of 3 is irrational

2. Originally Posted by teramaries
prove log base 2 of 3 is irrational
$\displaystyle \log_23=\frac ab \implies 2^{a/b}=3\implies 2^a=3^b$

Now look at the prime factorization of both sides and see this forces $\displaystyle a=b=0$ which is absurd in our case.

3. Originally Posted by teramaries
prove log base 2 of 3 is irrational

Assume $\displaystyle \log_2 3 = \frac{p}{q}$ where p and q are co-prime positive whole numbers

$\displaystyle \Rightarrow 3^q = 2^p$.

Now consider the last digits of $\displaystyle 3^q$ and $\displaystyle 2^p$ to get the required contradiction.

4. A slight variation of previous proofs . . .

Assume $\displaystyle \log_23 = \frac{p}{q}$ where $\displaystyle p$ and $\displaystyle q$ are coprime positive integers.

Then: .$\displaystyle 2^{\frac{p}{q}} \:=\:3 \quad\Rightarrow\quad 2^p \:=\:3^q$

Since $\displaystyle p$ is a positive integer, $\displaystyle 2^p$ is even.

Since $\displaystyle q$ is a positive integer, $\displaystyle 3^q$ is odd.

Hence: .$\displaystyle 2^p \:\ne\:3^q$

. . Q.E.D.