Wow, I'm really behind on posting articles. Here's one from last month. I think the title scared everybody away:

What Are Logarithms?

http://www.livescience.com/50940-logarithms.html

I've been outdone on how to to explain logarithms simply to a rough approximation. From How Not to Be Wrong by Jordan Ellenberg (Chapter 11 I think).

The logarithm of a postiive number N, called log N, is the number of digits it has.
Wait, really? That's it?
No. That's not really it. We can call the number of digits the "fake logarithm," or flogarithm. It's close enough to the real thing to give the general idea of what the logarithm means in a context like this one.