Logarithms are just orders of magnitude with a glow up, part 2

This post continues the discussion from part 1, in which I attempt to explain the concept of logarithms to myself, in the guise of explaining it to a novice. As before, this is primarily for my entertainment and edification; other people will either not need the explanation or can find better ones elsewhere.

(You’re still reading this?! OK, I guess I can’t stop you, but don’t say you weren’t warned.)

Notations

So far I’ve discussed orders of magnitude, including negative and fractional orders of magnitude, and defined a logarithm of a number as the (possibly negative and/or fractional) order of magnitude corresponding to that number. Now it’s time to simplify the discussion by introducing some basic mathematical notation.¹

Using mathematical notation tends to freak some people out, because they see it as an entirely separate language which they don’t and can’t understand — like staring at text in Japanese if you don’t know any of the characters. But it’s better thought of as simply a shorthand way to express things that would take longer and be more cumbersome to express in conventional prose. Thus, for example, it’s easier to write 2 + 3 = 5 than it is to write “two plus three equals five,” and easier to write 10×10×10×10 or 10⋅10⋅10⋅10 (my preferred notation) than “ten times ten times ten times ten.”

With that in mind, let’s introduce the following notation: We represent a one order of magnitude increase as being multiplied by 10¹ = 10, a two order of magnitude increase as being multiplied by 10² = 10⋅10 = 100, a three order of magnitude increase as being multiplied by 10³ = 10⋅10⋅10 = 1,000, and so on. By convention we have 10⁰ = 1, i.e., a “zero order of magnitude” change or no change at all, as discussed previously.

Note that this notation makes it simple to see the effects of multiple increases by various orders of magnitude, using the addition rule discussed previously. For example, an increase of two orders of magnitude followed by an increase of three orders of magnitude is represented as multiplying by 10² ⋅10³ = 10²⁺³ = 10⁵ , or an increase by a factor of 10⋅10⋅10⋅10⋅10 = 100,000.

What about a one order of magnitude decrease? We represent that as being multiplied by 10^-1 = 1/10 = 0.1. Similarly, a two order of magnitude decrease is represented as being multiplied by 10^-2 = 1/(10⋅10) = 1/100 = 0.01, a three order of magnitude decrease as being multiplied by 10^-3 = 1/(10⋅10⋅10) = 1/1,000 = 0.001, and so on.

Again the effects of combined increases and decreases by various orders of magnitude can be found using the addition rule discussed previously. For example, an increase of two orders of magnitude followed by an decrease of three orders of magnitude is represented as multiplying by 10² ⋅10^-3 = 10^2-3 = 10^-1 , or an overall decrease by a factor of 10.

As a side note, an increase by, say, two orders of magnitude followed by a decrease by two orders of magnitude is represented as 10² ⋅10^-2 = 10^2-2 = 10⁰ . We previously adopted the convention 10⁰ = 1, and we see now that that makes sense, since in this case there was no overall change.

We also have 10^-2 = 1/100 = 1/(10² ). So a negative value for orders of magnitude (i.e., decreasing by one or more orders of magnitude) converts multiplication into division, as we’ve discussed previously. We also have 1/(10^-2 ) = 1/(1/100) = 100 = 10² , by symmetry.

What about fractional orders of magnitude? We can represent an increase by half an order of magnitude as being multiplied by 10^1/2 = 3.1623 (approximately). Again the addition rule for orders of magnitude can be used: an increase by half an order of magnitude followed by another increase by half an order of magnitude is represented as being multiplied by 10^1/2 ⋅ 10^1/2 = 10^{1/2 + 1/2} = 10¹ = 10, as we would expect from the previous discussion.

Finally, how do we represent the logarithm of a number? We use the word “log,” of course: log(10³ ) = 3, for example. Often the parentheses are omitted: log 10^-1 = -1.

Multiplication by addition

Now let’s talk about something that was very important historically but is almost a curiosity now. Suppose you don’t have access to a computer, a calculator, or even an abacus. How can you do calculations like multiplying larger numbers, like 16 times 126, without resorting to pen and paper?

Let’s go back to the examples of fractional orders of magnitude I used previously, that multiplying by a factor of 16 corresponds to an approximately 1.2 order of magnitude increase, and 126 corresponds to an approximately 2.1 order of magnitude increase. In our new notation we have 16 = 10^1.2 and 126 = 10^2.1 . Alternately, 1.2 = log(16) and 2.1 = log(126), using the notation for logarithms.

Let’s also get better values for the logarithms of 16 and 126. Fortunately, way back in 1624 someone compiled a table of logarithms of numbers from 1 to 20,000 and from 90,001 to 100,000; on page 35 we find that the logarithm of 16 is 1.2041 (to four digits) and on page 36 we find that the logarithm of 126 is 2.1004 (also to four digits).²

Using our new notation we then have 16⋅126 = 10^1.2041 ⋅10^2.1004 . But by our rule of adding orders of magnitude we have 10^1.2041 ⋅10^2.1004 = 10^{1.2041+2.1004} = 10^3.3045 .

So we are looking for a number X = 10^3.3045 ; this is equivalent to writing log(X) = 3.3045. We can look in the same table we used before to find a number whose logarithm is 3.3045, and on page 55 we find that that number is 2016. This is the value we are looking for, the number corresponding to 16 times 126.

Of course, it would be tedious to carry around a large book just to do multiplication. The solution was the slide rule. The following is a gross simplification of how it might work:

We have two straight rules made of wood, plastic, or metal, each with numbers marked on it, increasing left to right from one end to the other. For example, on one rule the number 16 is marked at a distance of 1.2 cm from the left end of the rule, and the number 2016 marked 3.3 cm from the left end. On the second rule the number 126 is marked at a distance 2.1 cm from the left end of that rule. In other words, in all three cases the distances in cm from the left end of the rule are the logarithms of the numbers in question.

If we line up the left end of the second rule with the number 16 on the first rule (i.e., 1.2 cm to the right of the left end of the first rule), on the second rule the number 126 will line up opposite the number 2016 on the first rule (since 1.2 cm + 2.1 cm = 3.3 cm). Thus we’ve found the result of the multiplication 16 times 126.

Of course, for such a rule to be useful it would need to have all numbers marked on it, and at the size mentioned would be too small to be useful. Actual slide rules used for multiplication contain only the numbers from 1 though 10, since any multiplication with larger numbers can be done by fiddling with the decimal places. For example, we have 16⋅126 = (1.6⋅10)⋅(1.26⋅100) = (1.6⋅1.26)⋅(10⋅100) = (1.6⋅1.26)⋅1,000. So we really only need to multiply 1.6 times 1.26, and then multiply the resulting value by 1,000.

This concludes the second post. I may or may not make more posts in this series.