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$\therefore log_2(3)$ is not a rational number. Proof by contradiction.    \item Show that $\log_{\sqrt{2}}(3)$ is not a rational number  \\  Assume that $\log_\sqrt{2} (3) = \frac{p}{q}$ where p, q are integers.   \[\sqrt{2}^{\log_{\sqrt{2}} (3)} = 3 \]  \[ \sqrt{2}^\frac{p}{q} = 3 \]