## Tuesday, July 16, 2013

### A Simple Refresher of Logarithms

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 102 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 103 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 101.1761 = 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 10x button and wa-lah!    0.500 is calculated which is the  "antiLOG" of -0.3010.

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 =  100.8751 = 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.   101.4771= 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  e2.3026= 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.