prove log base 2 of 3 is irrational

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- Jun 30th 2010, 04:59 PMteramarieslog proof
prove log base 2 of 3 is irrational

- Jun 30th 2010, 06:18 PMchiph588@
- Jun 30th 2010, 06:21 PMmr fantastic
Use proof by contradiction.

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. - Jul 1st 2010, 05:42 AMSoroban

Avariation of previous proofs . . .*slight*

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.*