A little non-standard fun with Logs

Sorry this one is a little rushed, but I woke up today to see this post on Twitter:

@wahedahbug I'm with you. Anyone else wanna show Sadie and I why logs are actually cool and interesting?

— Casey Rutherford (@rutherfordcasey) April 16, 2014

By lucky coincidence I’ve just been talking about logs and exponentials with my older son.  By unlucky coincidence he’s been sick for a few days, but he was feeling well enough this morning to talk a little.  I thought it would be fun to try to answer the question on twitter with a few non-standard log examples for kids.

The goal was just to show a few examples that don’t appear to have anything whatsoever to do with logs.  I’m not trying to go into any detail, just wanted to show some fun and unusual math results.

The first bit of the talk was to switch from log base 10 to log base e.  The first result we talked about was that the area under the curve y = 1/x from 1 to x is equal to ln(x), followed by a couple of surprising results related to that:

The next part of the talk was about the distribution of prime numbers.   There are so many amazing results here that it was hard to even understand how to limit the talk to 5 minutes.  I ended up choosing three ideas – the approximate number of primes less than n, the approximate number of twin primes less than n, and the amazing Erdos-Kac theorem:

The last part of the talk was about the famous equation .  We’ve spent a little time in the past talking about this equation, so it wasn’t the first time he’s seen it.  The nugget I wanted to extract out of the equation this time around was that this equation allows us to talk about ln(-1).   After the video I explained some of the complications that arise when you talk about logs of negative numbers, but those details weren’t important for this talk.  For now, all I wanted to show is that ln(-1) might be equal to .

So, the nice little question on twitter made for a fun morning.  Not sure this is exactly the answer that anyone was looking for, or if it is even an answer at all, but it was neat to have these conversations with my son this morning.