Table of Contents

• Prequiz...
• Everything you always wanted to know about logs (but were too bored to ask)
• 1: When size matters
• 2: There must be a better way!
• 3: Special cases
• 4: The log is the power!
• 5: What about messy numbers?
• 6: And SMALL messy numbers?
• 7: Your turn with messy numbers
• 8: Antilogs
• So what are they good for?
• 9: Measuring earthquakes
• 10: Measuring acidity
• 11: Differences in acidity
• Review
• 12: Summary

Special Cases

We know that

log(1000) = 3, and

log (0.001) = -3

What about a number that has both a "large" part before the decimal and a "small" part after the decimal? Such as...

log(1000.001) = ???

Remember, the whole purpose of logs is to tell you approximately how big the number is. Therefore, the decimal part (0.001) is really pretty unimportant compared to the whole-number part (1000). So, the log of this number is pretty close to the log of 1000, plus a tiny bit thrown in for the 0.001. In fact, with the help of Google, I can find out that

log(1000.001) = 3.00000043

All right, one more special case...

Somebody out there is asking, what about negative numbers? Where do they fit into all of this??? Well, remember the graphic below? It outlines all the numbers we can take logs of. There are very big numbers (even bigger than 100,000), and very small numbers (smaller than 0.0000001), but there are no negative numbers.

The short answer is, we can't take the log of a negative number.

...which is generally all right. Usually we are using logs because we want to compare the amount of something we counted ... and that's usually a positive number, otherwise we couldn't count it.

Sometimes it does cause problems (especially in statistics), so people who use logs a lot have ways to get around those problems, but we won't discuss that here. For our purposes, just remember you can't take the log of a negative number, even if you do feel sorry for it.