Logarithms (aka Why everyone hates John Napier!)

Math Prerequisites: Log, Semi Log

Logarithms, developed in the early 1800’s by John Napier in Scotland, provide a convenient means of plotting values that range over several orders of magnitude (an order of magnitude is a power of 10).  Logs may be transcribed onto movable sticks to ease the task of multiplying and dividing since log(A) + log(B) = log (AB).  In fact, to ease the tedious task of multiplying and dividing was John Napier’s original motivation as he found arithmetic annoying.

In this exercise, you will create semi-log graph paper. Semi-log means that the vertical or y axis is scaled to the log but the horizontal or x axis is linear.

Step 1: Calculate the log of the numbers from 1 to 10. It is far more instructive to guess the power of 10, or exponent, that will give the correct answer than to simply use the log function on your calculator.  For example: to find the log of 2, try an exponent of 0.1.  10 raised to a power of 0.1 ie 10 ^ 0.1 = 1.25 which is <2 so try a larger exponent.  Try an exponent of 0.3 ,  1-^0.3 = 1.99 which is very close to 2.

Fill in the rest of the blanks:

number 1  2  3  4  5  6  7  8  9 10
exponent 0  0.3  0.48  0.778 1
10^exponent 1  1.99 10

 

Now plot these points on the linear – linear graph below with exponent on the vertical (y) axis.  Draw a horizontal line for each exponent on label that line with the corresponding number.   The x axis represents the x value of a function, and the y axis will be the log of the function.

Graph - 1 by 10

Voila! You have just developed your own semi-log graph paper.

SLIDE RULE:

John Napier, developed a method of multiplying numbers by adding their logarithms. He made scales or rulers similar to the semilog “y” axis on your graph and scribed these on sticks of wood.  Being Scottish and somewhat dramatic, he called these sticks “logs”.   Combining the words “log” and “arithmetic” he named this system “logarithms”.  To make your own “logarithm sticks”, print out the semi-log diagram or draw a similar graph on a piece of graph paper.  Cut the graph into 2 vertical strips being sure to label each of the horizontal lines on each strip.

Now multiply 2 x 2. Simply line up the papers next to each other so the line you labeled “1” is next to the line labeled “2” on the next strip. Now find the line labeled “2” on the second strip and read across to the corresponding line on the first strip.  It should read “4”.  Try reading the value for the line labeled “3”, the answer should be “6”,

Conclusion, you can multiply two numbers simply by adding their “logs”.

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