I didn’t feel very well today, either. I read some of Infinite Powers and worked on my logarithm problem. I am not ready to present a proof, but I have satisfied myself that if $a$ and $b$ are natural numbers greater than $1$, then they must be powers of the same base in order for $\log_a b$ to be rational. I’m not sure how to begin thinking about other types of numbers, and it’s possible I will lay that question aside for now.1
I am still not making much progress on my calculus review. It’s an important part of this project, because I hope it will prepare me to earn some money tutoring. The truth is, though, that a lot of the time I am just not hale of mind. I’m doing my best.
- To do. ↩︎
Prayers for better days ahead.
Of course you are doing your best, and what you are doing is delightful. Please be gentle with yourself.
It is not completely clear to me that there is a place to go with your logarithm problem beyond where you are. You’re hoping to simplify an equation that is already not very complicated. Don’t feel like you have to give up because that’s my thought after a couple minutes reflection, but also don’t feel like there is any shame in deciding that you’ve extracted what there is to extract from the problem. And while you’re at it, please be gentle with yourself.
I’m working on being gentle with myself. It is, as you know, something of a difficult trick. I agree that there might not be an interesting answer for non-integers. I’m curious about negative numbers, though.