A math odyssey
I had a lot of anxiety today, which didn’t lend itself to study. I did a little work on the logarithm question I’ve been talking about, but nothing else.
(For those wondering, the question is what the relationship between $a$ and $b$ must be in order for $\log_a b$ to be rational. It arose from a exercise in Analysis with an Introduction to Proof that asked me to prove that $\log_2 7$ is irrational.)
What a wonderful question, about logs!
I was initially worried, because there is a question that looks similar but that is quite difficult. The Gelfond-Schneider Theorem says that if $a$ and $b$ are algebraic and $b$ is not real and rational, then $a^b$ is transcendental. This was proven in 1934, and answers Hilbert’s 7th Problem.
Happily, your question isn’t equivalent to Hilbert’s 7th Problem. That doesn’t mean I know how to do it – it’s your problem, not mine – but it does mean I don’t need to instantly say, “Consider the possibility that if Hilbert couldn’t do it, then it might be fairly hard.” 😀
I’m glad I’m not trying to do one of Hilbert’s problems. Maybe someday.
[…] advice on Reddit about how to choose textbook exercises, and started writing up my work on the logarithm problem I brought up a while ago. Unfortunately, the last of those tasks proved quite fiddly, and I […]
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