Please find attached problem and my work.

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- Nov 30th 2010, 09:28 PM #1

- Nov 30th 2010, 11:12 PM #2

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- Dec 1st 2010, 12:23 PM #3

- Dec 1st 2010, 01:27 PM #4

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Proof by contradiction. Assume it is rational.

$\displaystyle \log_{8} 6 = \frac{a}{b}$

$\displaystyle 8^{\frac{a}{b}} = 6$

$\displaystyle 8^a = 6^b$

$\displaystyle 2^{3a} = 2^b*3^b$

$\displaystyle 2^{3a-b} = 3^b$

From unique factorization, this is impossible - a contradiction.

QED.