# GL(s,R)

## July 4, 2010

### Travel Log

Filed under: Uncategorized — Adam Glesser @ 9:52 pm
Tags: ,

Hey everyone,

I’m back from Europe and simultaneously elated, bummed, excited, nervous, stressed and a little sleepy. I had a productive trip and I can’t wait to tell you all about it. First, though, I need to turn the title into a pun.

Just kidding, the real pun is what follows:

A Question

Let me fast-forward to the second half of my trip. My roommate in Lausanne, the Korean dynamo Sejong Park, gave me the following problem (suitable for high school math competitions) on the first day. It took me a mere 10 days to solve it (although, to be fair, I only worked on it about 3 hours per day).

Decide which of the following quantities is greater:

$\dfrac{\log(5)}{\log(3)}$ or $\dfrac{\log(3)}{\log(2)}$

where the base of the $\log$ is fixed, but arbitrary. Try to work on it before you read the answer. If you don’t get it in the first 10 minutes, don’t feel bad, neither did any of the mathematics professors I asked (however, Romanian super-stud Radu Stancu apparently solved this in two minutes).

Oxford

Starting from the moment I walked through passport control in Heathrow airport and saw some bizarre parabolic light fixtures, I wished I had my camera. One of the highlights was getting to eat dinner at the Christ Church High Table (those of you into obscure literature and cult films might recognize the room as being where a certain H. Potter sups. Note that, unlike in the film, the room does have a ceiling). A combination of jet lag and hay fever made the first few days a bit of a blur, but eventually my host, David Craven, and I got down to work and watched Team America: World Police. After that, we finished the paper that just wouldn’t die and then I worked a bit on editing his new book on fusion systems.

Lausanne

A week in Oxford and then we took off for Switzerland. Thanks to a clever disguise and some misdirection, I was able to get into the country despite asking earlier in the week if Switzerland had been named after the font, Helvetica. Anyhow, the conference had everything: talks, night club dance-offs, barbecues, workshops, water closets, fencing, fighting, torture, revenge, giants, monsters, chases, escapes, true love, miracles… Simply put, it rocked. I got to hear about biset functors from Serge Bouc (and who better to hear them from, the guy wrote a Bouc on the subject) and learned the correct technique for pouring beer from the tap (as a lifelong teetotaler, this wasn’t a skill I had picked up before). Good things came to an end, eventually. I started missing my family and was worried about my growing (more beautiful by the day) wife  who is now 23 days and counting until the due date of the yet-to-be-named baby number 3 (I already suggested Glesser Escher Bach and my wife said, “no”). When I started thinking about how I could use $\{\infty, -\infty \} \cup \mathbb{Z}$ to demonstrate a non-associative binary operation to my older son, I knew it was time to come home.

Back home

So, I finished a paper and have between one and three new collaborations started. This might be a great year for research. However, my homecoming was not all roses. For starters, I see that while I was gone, Mr. Cornally set up and closed up shop on the beta version of his Standard Based Gradebook. Major bummer that I missed this. Eighty-seven blogs posts to go and I’m caught up on the math edublog section of my reader. What else did I miss (is that a cool new background over at $f(t)$)?

Coming soon

I wrote down several posts (or at least the outline of several posts) while on my trip. Some are pedagogical, others are more of those page view killers where I pretend to know something about something. In any case, they should be out reasonably soon, unless, that is, I preempt them with…

Coming sooner

I start teaching a summer precalculus course this week if enrollment is high enough. This is going to be the official start of my SBG experience. I finished crafting the syllabus today and tomorrow will finalize the course standards; by finalize, of course I mean that they are still subject to change as soon as I figure out what I’m doing. I’ve decided to split the grade up as 80% for standards and 20% for the final. As someone who always wanted to keep the final worth at least 40%, I’m still a bit wary of taking it down to Cornally levels. I also decided to toss out the midterm since the course is only 6 weeks long. As soon as I finish my standards list, I’ll post them here for your convenience and, hopefully, your criticism.

The picture below is only inserted to stop you from seeing the answer to the Travel Log problem

Okay, here it comes…

If you’ve struggled long enough and would like to see the answer, here it is:  $\dfrac{\log(3)}{\log(2)} > \dfrac{\log(5)}{\log(3)}$.

What? You say you had figured that out already by plugging it into a calculator? Well that isn’t very sporting, is it. Anyway, one can deduce the above inequality without resorting to newfangled technology. Let’s make a couple of observations. First, using the change of base formula, we can rewrite the two fractions:

$\dfrac{\log(5)}{\log(3)} = \log_3(5)$ and $\dfrac{\log(3)}{\log(2)} = \log_2(3)$.

Clearly, both of these are greater than $1$ and less that $2$. Let’s make some room by multiplying them all by $2$. We then get $2\log_3(5) = \log_3(25) < 3$ and $2\log_2(3) = \log_2(9) > 3$ and we’re done. The general idea here is that if we can bound two logs by the same integers, then multiplying by a suitable positive number, we can tell the logs apart. Simple when you see it!

## 4 Comments »

1. My solution was a couple-minutes one, but I assure you it was a freak accident. I’m pretty bad at math competitions.

3^(3/2) is 27^(1/2) so it has to be less than 5, so log(5)/log(3) is less than 3/2.

2^(3/2) is 8^(1/2) which is less than 3, so log (3)/log(2) is greater than 3/2.

Comment by Jason Dyer — July 6, 2010 @ 9:22 am

• er, first one read “greater than 5”

Comment by Jason Dyer — July 6, 2010 @ 9:23 am

2. Jason, my hat goes off to you. The way I eventually solved it was much closer to your method than the one I presented above (although I did a lot more work to get where you did). These math competition style problems are so fun, but I find them far more challenging than most of the research I do!

Comment by Adam Glesser — July 8, 2010 @ 8:15 am

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