Some folks are intimidated by the mathematics of
modeling. I am here to tell you that by
taking a little time to really understand a few basic concepts in math (which I
am going to help you with) you are ready to approach and understand the
mathematical underpinning of many models.

When and where many of us went to school, math was not very well
taught. Indeed, many of us were taught
to manipulate numbers to get answers but we were not schooled in exactly what
the process meant in a larger sense. We
soldiered through math course with a good portion of us hating it but never really seeing what it
was good for. In this blog I am hoping
to refresh our understanding of logarithms so that we can clearly understand and use them in models.

The first thing to understand about logarithms is that they are just
another way of expressing a number. All
logarithms have a BASE. The one we see
much of the time is BASE 10. So the “power” or exponent that 10 is raised
to is its logarithm or log. So the log
of 100 is 2 since base 10

^{2}is equal to 100. The base10 log of 10 is equal to 1 for the same reason. Similarly, if we know that the base 10 log is 3 we know the number is 10^{3}or 1000. Calculating logs are easy for simple integers but considerably more difficult for those numbers in between. In the old days before calculators we had “log tables” to give us these numbers. Today we just put the number in our calculator; for example, enter 15 and hit LOG to get the log of 15 or 1.1761. Thus 10^{1.1761 }= 15.
Numbers less than 1 have negative
logs. Log of 0.5 = -0.3010
or 10

^{-0.3010}= 0.5.
Try putting in -0.3010 for x and hit the 10

^{x}button and wa-*! 0.500 is calculated which is the "antiLOG" of -0.3010.***lah**
Logs are handy in that adding the logs of numbers is the same as
multiplying the numbers. Add -0.3010
to 1.1761 = 0.8751. The antiLOG of
0.8751 = 10

^{0.8751}= 7.5 or the same as 0.5 x 15. Similarly subtracting logs is the same as dividing the numbers. Subtract -0.3010 from 1.1761 = 1.4771. 10^{1.4771}= 30 = 15/0.5. Flip it around and Subtract 1.1761 from -0.3010 = -1.4771. 10^{-1.47711}= 0.03333 or the same as 0.5/15. For us folks who have been around since before calculations, this is how a slide-rule works because it has logarithmically-spaced scales, you are adding or subtracting logs by sliding the scales to do multiplication or division on the slide-rule.
You will see a lot of logs expressed as base 10 because that is our
basic number system (a system based on 12 might be better) and that is how many
fingers we have. Logarithmic scales are convenient for
compressing large ranges of numbers because in base10 the distance between each
integer represents 10x. For example, the difference in shaking
amplitude between a Richter magnitude 7 earthquake is 10 times higher than a magnitude
6 because the scale is base 10 logarithmic.

Now that we understand algorithms
in base 10 it is time to understand logarithms with the NATURAL base number. This natural BASE number is 2.718281... or the italicized
lower case letter “e”. The only thing you need to know about “

*e” is*that it is an important number in mathematics and actually much more “natural” than 10. Thus it is called the NATURAL log (sometimes called LN or ln). The same rules apply, that is, the natural log of 10 is 2.3026 (it’s also on your calculator - sometimes with key named LN) which means that*e*^{2.3026}= 10.
Next week I am going to talk about the subject of integration and how
it fits into modeling. We are going to
see “

*e”*again and we will walk through a practical model using a tracer gas that lets us estimate the ventilation rate in any workplace room or any residential volume.
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