How do you use $\log_2(3)$ is irrational to provide a counterexample for: if a and b are irrational, then $a^b$ is irrational

Question

I have seen a similar proof having to do with $a=\sqrt{10}$ and $b=\log_{10}(4)$, which I thought was excellent because this equals $2$. But I am not sure how to apply this because I can not think of a rational root for $3$. What can be done here?

Answer 1

Taking your own similar counter proof example, you are given that $ \log_{10}(4) $ is irrational. You take another irrational number which comes from the base used in the logarithm notation, in this case $ 10 $.

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From this, if you have $ \log_2(3) $ as irrational, the second irrational number would be (base of give log value) $ \sqrt 2 $, and you have your counter proof:

$$ \begin{align} \left( \sqrt{2} \right)^{(c) \log_2{(3)}} &= \left( (2)^{\dfrac{1}{2}} \right)^{(c)\log_2(3)} \tag{assume c is any integer } \\ &= \left( 3^{\dfrac{c}{2}} \right)^{log_2(2)} \\ &= \left( \sqrt{3^c} \right)^1 \end{align} $$

From this, you can see that the value is rational for any even valued integer $c$.