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Prove by contradiction that $\log_5 8$ is irrational. While I understand that this is true, I am struggling to prove it by contradiction. Thank you for any help!

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closed as off-topic by John B, Silvia Ghinassi, choco_addicted, colormegone, Harish Chandra Rajpoot Mar 16 '16 at 3:47

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  • $\begingroup$ Now I wonder if there is a simple proof for statements like this that is not by contradiction ... I no longer wonder this $\endgroup$ – Wouter Mar 15 '16 at 15:43
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By contradiction, so suppose it is rational and that it can be written as $p/q$ (in its simplest form). But then: $$\log_5 8 = \frac{p}{q} \Leftrightarrow 5^{\frac{p}{q}} = 8$$ Rewrite this as: $$5^p = 8^q$$ Hint: what can you say about both sides being odd or even?

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  • $\begingroup$ THis leads to a contradiction by even and odd and by Fundamental Theorem of Arithmetic as well. $\endgroup$ – user74489 Mar 15 '16 at 15:45
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Suppose that $\log_{5}8=p/q$

$$\log_{5}8=p/q$$

From $\log$ rules:

$$\iff 5^{p/q}=8$$

Power of $q$ for both sides:

$$\iff 5^{p}=8^q$$

The RHS is even and the LHS is odd and it is can't be

$\square$

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  • $\begingroup$ Thank you, that makes complete sense! $\endgroup$ – Pamela Graham Mar 15 '16 at 20:28

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