MathBench > Measurement

Logs and pH

What about messy numbers?

So far we have been finding logs of pretty easy numbers -- various arrangements of a '1' and some zeros. So, it's easy to figure out that log(100) = 2 and log(1000) = 3. But of course (being scientists) we know that the numbers we're really interested in aren't going to be so simple.

measuring a logHow about the log of 180? 279? 736.2? Well, first of all, we know that all of the logs for these numbers must be between 2 and 3. But where exactly? There is no easy formula for calculating this ... here is a case where you really do need a calculator (or a sliderule, or a lookup table, or enough time to do a little calculus).

But, you can certainly make an informed ESTIMATE of the log.

For example, 180 is about a tenth of the way between 100 and 1000. So the log should be somewhere around 2.1 . Let's check that out...

log(180) = (We guessed 2.1)

Well, that's more or less in the ballpark. If you have done any log-transforming of graphs, you know that logs tend to exagerate small differences (or "unsquish" data). So it makes sense that the log would be a little higher than our original guess.

Let's try another one. 279 is about a third of the way between 100 and 1000. We could guess 2.25, but let's increase that a little ("unsquish" it). How about 2.35? (Feel free to make your own guess here...)

log(279) = (I guessed 2.35)

Well, as you can see, estimating logs takes a bit of practice... How about 736.2? This is about 2/3 of the way between 100 and 1000, and I'm gonna make sure I unsquish a lot here ... how about 2.85?

log(736.2) = (I guessed 2.85)