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.
How 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)
Whew!
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