We usually think about numbers using a linear scale. That is, we usually think of the number line that is constructed with the integers spaced a constant unit apart. Once we know the unit length (the distance from 0 to 1), then we use the same length to reach 2 from 1 and then to reach 3 from 2 and so forth. Negative integers, which are the additive inverses of the positive integers, can be marked by counting the same unit distance in the reverse direction. Rational numbers represented as \(p/q\) (with integers \(p\) and \(q > 0\)) can be constructed by subdividing the unit length into \(q\) equal parts and then counting \(p\) such subunits (going left if \(p<0\)).
The linear scale is convenient for addition. If we want to add two numbers \(a\) and \(b\), then we can identify their locations on the number line and construct a vector (arrow) corresponding to each \(\vec a\) and \(\vec b\) with the tail at \(0\) and the tip at \(a\) and \(b\), respectively. Addition \(a+b\) corresponds to adding the vectors, with \(\vec a\) placed with the tail at \(0\) and then \(\vec b\) placed with its tail at the tip of \(\vec a\). The resulting location of the tip of \(\vec b\) will be at the number corresponding to the value of \(a+b\).
The vector interpretation of addition on a linear scale is illustrated in the interactive demo below. You can control/choose the value of \(a\) and \(b\) by dragging the red (\(a\)) and blue (\(b\)) points. The vector addition is then illustrated immediately below. Try a variety of addition facts by adjusting the values of \(a\) and \(b\). Be sure to include some examples involving negative numbers (one or both values).
A logarithmic scale is designed to be convenient for multiplication. Each positive number will be represented by vector of some length, called the logarithm of the number, such that when the vectors are added the result corresponds to the product. That is, if \(a,b > 0\), then there are vectors \(\log(a)\) and \(\log(b)\) so that \[\log(a) + \log(b) = \log(a \cdot b).\]
The dynamic graph below illustrates the logarithmic scale. You can drag the locations of \(a\) and \(b\) to set your two numbers. Notice that the vector sum always corresponds to the logarithm of the product of numbers associated with \(a\) and \(b\).
Set \(a=2\) and then move \(b\). Notice that when \(b=3\), the vector sum is at \(ab=6\). The spacing of numbers is exactly designed so that every time you move to the right by the vector of length \(\log(a)=\log(2)\) the numbers exactly double.
In a similar way, if you set \(a=3\) and move \(b\), you see that the values \(b\) and \(3b\) are always separated by \(\log(a)=\log(3)\). The logarithmic scale captures all such relations.
What happens if you put \(a=b\) (at the same location)?
What is the length of the vector associated with \(a=1\)? Why?
What do you notice about the spacing between 1, 2, 3 and 4 in comparison to the spacing between 10, 20, 30 and 40? Why?
If we were to extend the line further, where would we have 200, 300, 400 and 500? Where we would put numbers less than 1?