Logarithms

If you're like most people your first encounter with math was to learn how to count, you know, whole numbers 1, 2, 3,...etc. That wasn't so bad. Sometime later you learned how to add and subtract these numbers - again, no problem. However, about the time you entered third grade things started to get complicated - there was multiplication, division, fractions, remainders, times-tables, and more. Sure - now that you've memorized the times-table you don't have to add 8 nine times to know that 9 times 8 is 72. Still, if you're like most people, you generally find it easier to add and subtract numbers than to take their product and quotients. If only there were some system whereby we could reap all the benefits of multiplication and division without all the work? Well there is such a system, and all that's required to use it is to learn a new way of counting. In this system the counting numbers 1,2,3,...etc. actually represent factors of 10. That is, 2 means 10X10 or 100, 3 means 10X10X10 or 1000, 4 means 10X10X10X10 or 10000, and so on. The numbers in this new counting system are referred to as logs. The table below shows the relationship between the log values (right-most column) and the large numbers they refer to (left-most column). The right-most column is referred to as an interval scale because the interval (difference) between successive numbers is constant. The left-most column of numbers is referred to as a ratio scale because the ratio between successive numbers is constant.

                            Ratio to Interval Scale  

                                    
  Number          As Product          As Power           As Log              


100,000.            10X10X10X10X10             105                  5           
10,000.             10X10X10X10                104                  4           
1,000.              10X10X10                   103                  3           
100.                10X10                      102                  2           
10.                 10                         101                  1           
1.                  1                          100                  0           
0.1                 1/10                       10-1                 -1          
0.01                1/(10X10)                  10-2                 -2          
0.001               1/(10X10X10)               10-3                 -3          
0.0001              1/(10X10X10X10)            10-4                 -4          
0.00001             1/(10X10X10X10X10)         10-5                 -5

Now here's the good part. If we need to know the product of two large numbers, say 1000 and 100,000, we don't have to multiply. In our new counting system, the product of these two numbers is represented by the sum of their log values. The log of 1000 is 3, the log of 100,000 is 5, and so the log of 1000 times 100,000 is 3+5 or 8. The same is true for division. Express the ratio 10/10,000 as a log value. The log of 10 is 1, the log of 1/10,000 is -4, adding the log values yields -3 which is the log of 10/10,000.

OK - OK all this is very simple you say - but does this mean that every number we encounter will be a factor of 10? The answer is of course no - but this poses no real problem. We know that a factor of 10 corresponds to 1 log unit. Any other factor less than 10 (i.e. 2, 3, 4, etc.) must therefore correspond to some fraction of a log unit. For our purposes we need only remember one of these values. We will remember that a factor of 2 corresponds to one-third of a log unit, that is log(2) = 0.3. What is the log of 20,000? Well, 20,000 is 2X10,000. The log of 2 is 0.3 and the log of 10,000 is 4. Therefore the log of 20,000 is 0.3+4 or 4.3.

Laws of Logarithms

i. log(xy) = log(x) + log(y)

ii. log(x/y) = log(x) - log(y)

iii. log(xn) = nlog(x)

iv. log(0) is undefined