Thursday, December 08, 2005

Double, double, toil and trouble

A few people (i.e. non-math-geeks) have asked me what the title of this blog, Log Base 2, means. In mathematics, log base 2 is a logarithm function. It's actually not too hard to understand, and I have a nice example below that I hope will clarify it.

The growth of blogging in the last 3 years has been dramatic, as shown in this figure from Technorati, a company which tracks blogs:


This kind of growth is often described as "exponential". Exponential growth means that the rate of growth is proportional to the current size, so it starts out slowly but soon grows very quickly. The figure notes a "doubling in size approx. every 5 months". If we let N0 be the number of blogs in March of 2003 (roughly 100,000), and N(m) be the number of blogs m months later, then we can write this as
N(m) = 2m/5N0

So, after 5 months, we have

N(5) = 21N0 = 2N0


and after 10 months, we have

N(10) = 22N0 = 4N0


and after 15 months, we have

N(15) = 23N0 = 8N0


and so on. Which brings us to log base 2, or log2 as it's usually written.

The log2 of a number means the power of 2 that gives that number. So

log2 2 = 1 because 21 = 2


and

log2 4 = 2 because 22 = 4


and

log2 8 = 3 because 23 = 8


Let's try redrawing the figure using log2 of the number of blogs (expressed in hundreds of thousands):



An increase of 1 unit corresponds to a doubling of the number of blogs. If you look at the start of 2004, the curve is at roughly 4, and by the end of 2004, it's at 6. This is an increase of 2 units, meaning a quadrupling of the number of blogs in 12 months (that is, a doubling every 6 months, not 5). At the start of 2003, the curve is steeper, meaning an even more rapid growth. So the Technorati figure wasn't quite right in saying there was "consistent doubling over the last 36 months."

Although the log base 2 units are very convenient for working out the doubling time, they hide the true numbers. So it's convenient to re-label the graph:


This is called a logarithmic scale. It takes a little getting used to, but it's very good at showing relative changes for quantities that vary over several orders of magnitude.

There! Clear as mud? Log base 2 is particularly useful in computer science because computers use binary (base 2). And in addition to being a bit of a math geek, I'm also a bit of a computer geek, and, well you get the point ... (and the name was a pun too, of course).
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8 Comments:

Blogger Raywat Deonandan said...

"in addition to being a bit of a math geek, I'm also a bit of a computer geek."

You renaissance man, you!


;-)

12:01 PM, December 09, 2005  
Blogger Nick Barrowman said...

This above all: to thine own self be true,
And it must follow, as the night the day,
Thou cans't not be false to any man

10:53 AM, December 10, 2005  
Anonymous Joe Dawson said...

It's actually not too hard to understand, and I have a nice example below that I hope will clarify it.

Reading this made me think of a HP ink cartridge I had in the 1990s. On this two inch square black and green cartridge it said “DO NOT EAT”. I keep thinking who would mistake this for a food and try to eat it and was able to read the text “DO NOT EAT” and understand it? In short who was the person this message was targeted for?

More then ten years later I have never been able to think of one potential example where anyone would have been saved by reading this warning text.

This blog is not for the faint of mind or the faint of mathematical skills. Based on past postings to the blog if you really don’t understand what log base 2 means, your likely not likely the target market for this blog.

The explanation while quite good and quite complete was not for the mathematically challenged.

I am glad you called it a nice example and not a simple one. 

All that being said, keep up the interesting comments and postings.

All the best,

Joe

11:44 AM, December 11, 2005  
Blogger Zeno said...

Nice intro for the non-mathphobic, Nick.

Say, what did you use to create and embed the equations?

3:06 PM, December 11, 2005  
Blogger Nick Barrowman said...

Thanks, Zeno. (I wonder if there's any point trying to explain it for the mathphobic?)

I did it in HTML, using the <sup;> and <sub;> tags for super- and subscripts. Wikipedia has support for LaTeX, which would be wonderful (sigh!) ...

3:57 PM, December 11, 2005  
Blogger Nick Barrowman said...

Aargh, ignore those semicolons (after sup and sub). I was trying to render the closing angle brackets, but obviously didn't get it quite right.

3:58 PM, December 11, 2005  
Anonymous Anonymous said...

i wonder if theres any point of this comment about seven years later but the lizard/ chameleon? on every page is very freaky. you should get rid of it, otherwise the example was really well explained!

11:50 AM, January 11, 2012  
Anonymous roman said...

i like the lisard

7:41 PM, May 09, 2014  

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