# log proof

• Jun 30th 2010, 05:59 PM
teramaries
log proof
prove log base 2 of 3 is irrational
• Jun 30th 2010, 07:18 PM
chiph588@
Quote:

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

$\log_23=\frac ab \implies 2^{a/b}=3\implies 2^a=3^b$

Now look at the prime factorization of both sides and see this forces $a=b=0$ which is absurd in our case.
• Jun 30th 2010, 07:21 PM
mr fantastic
Quote:

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

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

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

Now consider the last digits of $3^q$ and $2^p$ to get the required contradiction.
• Jul 1st 2010, 06:42 AM
Soroban

A slight variation of previous proofs . . .

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

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

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

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

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

. . Q.E.D.