The purpose of this posting is to describe a mixed system of logarithmic units.
By the end of the posting, with just the use of a table of 19 (easily memorized) values, you will be able to find arbitrary powers and roots of arbitrary numbers, generally to within one percent accuracy.
This system could, in principle, have been invented a couple of hundred years ago. It is contructed mostly by stitching together a large variety of preexisting pieces: decades, decibels, R-series prefered numbers, pounds-shillings-pence money, order-of-magnitude arguments, percentage tweaking calculations.
My particular contribution here is putting together a mixed system of logarithmic units. I have never seen that done anywhere else. The system particularly exploits the numerical coincidence that
`qquadqquadqquad10^{1//240} \approx 1.01`,
where
- `10` is the base of the decimal number system we use,
- `1.01` is the factor corresponding to a tweak of one percent, (percent being the usual unit of tweaking arguments,) and
- `240` is the number of pence in a pound, a highly composite number.
An earlier posting defined the (well-known) logarithmic units decade, neper, and octave as `"lg"(10)`, `"lg"("e")`, and `"lg"(2)`, respectively. All logarithmic units in common use are at most simple rational multiples of one of these three units. So, there are decadic units, neperic units, and octavic (or binary) units.
First, let's define three decadic units, and then show why they are particularly useful as a system.
These logarithmic units are the decade (dec), the decibel (dB), and the point (pt). Let's defer discussion of the merits of these particular names.
The old British Imperial currency, its lineage dating back past Charlemagne, had a pound equal to twenty shillings, and a shilling equal to twelve pence.
These logarithmic units share the same arithmetic. The decade is twenty decibels, and the decibel is twelve points. Thus
- `1" decade" = "lg"(10)`,
- `1" decibel" = 1/20 "lg"(10) = "lg"(10^{1//20}) \approx "lg"(1.12)`, and
- `1" point" = 1/240 "lg"(10) = "lg"(10^{1//240}) \approx "lg"(1.01)`.
Arguments that use round powers of ten are a commonplace in technical disciplines. They are known as order-of-magnitude arguments, and people use expressions like "three orders of magnitude" to mean "very roughly a factor of a thousand". The decade is a precise order of magnitude. Order of magnitude arguments are, at root, logarithmic arguments.
The common logarithm, the one found by the 'log' key on a calculator, is the number of decades in the multiplying strength of a number.
The decibel is the multiplying strength of the twentieth root of ten, which is about `1.12`. Integer powers of this, i.e. numbers of the form `10^{n//20}` for `n` an integer, constitute the R20 series of prefered numbers, as now maintained in an international standard by the ISO. The 'R' in R20 honours the French engineer Charles Renard, the inventor of this system. The `20` tells us that we go from one power of ten to the next in twenty steps. These steps---this is Renard's invention---go up in constant proportion.
All of the values in stripy multiplication table
`\qquad\qquad\qquad[(1,1.78,3.16,5.62,10),(1.78,3.16,5.62,10,17.8),(3.16,5.62,10,17.8,31.6),(5.62,10,17.8,31.6,56.2),(10,17.8,31.6,56.2,100)]`.
come from what could reasonably be called the R4 series of prefered numbers. Interposing rows and columns, we can build a similar table for the R20 series. The R20 values between `1` and `10`, rounded to three significant figures, are the values in the second column. Their logarithms, expressed in decibels, appear in the first column ":
`\qquad\qquad\qquad{:(0" dB", \qquad1),(1" dB", \qquad1.12),(2" dB", \qquad1.26),(3" dB", \qquad1.41),(4" dB", \qquad1.58),(5" dB", \qquad1.78),(6" dB", \qquad2.00),(7" dB", \qquad2.24),(8" dB", \qquad2.51),(9" dB", \qquad2.82),(10" dB", \qquad3.16),(11" dB", \qquad3.55),(12" dB", \qquad3.98),(13" dB", \qquad4.47),(14" dB", \qquad5.01),(15" dB", \qquad5.62),(16" dB", \qquad6.31),(17" dB", \qquad7.08),(18" dB", \qquad7.94),(19" dB", \qquad8.91),(20" dB", \qquad10):}`
Put another way, if `n` is the numerical part of the logarithm in the first column, then the number in the second column is `10^{n//20}`.
In the seventh edition of the SI Handbook, published by the BIPM, the bel was defined as half of what we have been calling a decade. The decibel is a tenth of this, and so it is a twentieth of a decade. The eighth edition of the SI Handbook removed this definition of the bel, but did not supersede it.
The R20 values form a bridge between the two other kinds of units. Also, since, to three significant digits, the number 2 is an R20 value, the octave, `"lg"(2)`, is approximately a round number of decibels. This gives us a tie between octavic units and decadic units.
The point is the multiplying strength of the 240th root of ten, which is about 1.01. A change of one point is roughly equivalent to a change of one percent. It is also roughly equivalent to a change of one centineper, which gives a tie to neperic units.
Percentage tweaking arguments are widely used in technical disciplines. If a quantity goes up by three percent, say, then its square goes up by about six percent. If the length of a rectangle goes up five percent, while its width goes up by two percent, than its area increases by about seven percent. Addition of small percentage tweaks is used to do the work of multiplication---this is the hallmark of logarithms. Small percentage tweaks used in this way are being used as approximate logarithms.
This mixed system of logarithmic units is particularly well suited to calculation of approximate logarithms.
As an example, let's calculate the logarithm of `72900`. First, we rewrite this in reverse scientific notation, i.e. we factor it as a whole power of ten multiplied by a number between `1` and `10`:
`\qquad\qquad\qquad 72900 = 10^4 xx 7.29`.
Next, we break the second factor into two factors. The first factor is selected from the R20 values in the table. Since `7.29` is between `7.08` and `7.94`, we will pick one of these values. We usually pick the lower table value, unless our value is within about two percent of the higher value. Then
`\qquad\qquad\qquad 72900 = 10^4 xx 7.08 xx 7.29/7.08`.
We now calculate the ratio `7.29//7.08`. This could be done by long division, but we prefer instead the folowing :
`\qquad\qquad 7.29/7.08 = 729/708 = {708 + 21}/708 = 1 + 21/708.`
To get three or four digits of accuracy in the sum `1 + 21//708`, we need only one or two digits of accuracy in `21//708`. We can therefore afford to round the denominator, so that
`\qquad\qquad\qquad 21/708 \approx 21/700 = 3/100`.
Then `1 + 21//708 \approx 1.03`.
The same conclusion can be reached by making the proportion table :
`\qquad\qquad\qquad{:(708, 100%),(71, 10%),(7, 1%),(21, ?):}.`
The first row always consists of the target divisor or denominator, here 708, and 100%. One creates the second row by dividing through by ten or another simple divisor, until the value in the first column is comparable to or smaller than the target dividend or numerator, here 21. Numbers in the first column are usually rounded to the same number of decimal places (here zero) as in the first row. If the target numerator is a small multiple of the target numerator---here `21 = 3 xx 7,` this gives us what we are looking for, i.e. the desired percentage is `3 xx 1% = 3%.`
(We don't want a high multiple, especially if rounding has been severe. If, for instance, the numerator had been 51 instead of 21, then we would created yet another row by taking three times the `[7,1%]` row from the `[71,10%]` row and we would have had `[50,7%],` so that `51//71 ~~ 0.07.`)
Either way, here, we get
`\qquad\qquad\qquad 72900 = 10^4 xx 7.08 xx 1.03`.
From the table, we can rewrite `7.08 = 1.12^17`. The binomial theorem gives us that `(1 + x)^n \approx 1 + n x`, and so `1.03 \approx 1.01^3`. So
`\qquad\qquad\qquad 72900 \approx 10^4 xx 1.12^17 xx 1.01^3`.
Then
`\qquad\qquad\qquad "lg"(72900) \approx "lg"(10^4 xx 1.12^17 xx 1.01^3`)`,
`\qquad\qquad\qquad\qquad = "lg"(10^4) + "lg"(1.12^17) + "lg"(1.01^3`)`,
`\qquad\qquad\qquad\qquad = 4\cdot"lg"(10) + 17\cdot"lg"(1.12) + 3\cdot"lg"(1.01)`,
`\qquad\qquad\qquad\qquad = 4" dec" + 17" dB" + 3" pt"`.
Compound arithmetic with mixed units can be used for adding or subtracting the logarithms of different numbers, and even for multiplying or dividing by simple integers. These are the processes one needs when one is using logarithms to multiply, divide, or take simple roots or powers.
For more complicated powers, and often in other calculations, it more convenient to work in a consistent units. We can rewrite this mixed form using wiggle notation, and use that to calculate values in one or other of the given units, so that
`\qquad\qquad\qquad "lg"(72900) \approx 4;_{20}17_{12}3" dec"`,
`\qquad\qquad\qquad\qquad = 4_{20}17;_{12}3" dB"`,
`\qquad\qquad\qquad \qquad = 4_{20}17_{12}3;" pt"`.
For instance,
`\qquad\qquad\qquad 4;_{20}17_{12}3 = 4 +\frac{17 xx 12 + 3}{20 xx 12}`,
`\qquad\qquad\qquad\qquad = 4 +207/240`,
`\qquad\qquad\qquad\qquad = 4 +\frac{207 xx 1.04}{240 xx 1.04}`,
`\qquad\qquad\qquad\qquad \approx 4 + \frac{215}{250}`,
`\qquad\qquad\qquad\qquad = 4 +\frac{860}{1000}`,
`\qquad\qquad\qquad\qquad = 4.86`.
i.e. `"lg"(72900) \approx 4.86" dec"`, or `\log_{10}(72900) \approx 4.86`. This compares well with the more accurate value of `4.8627275`.
It is usually more convenient to work in points. The conversion doesn't require further division, at least here, and each point corresponds to a one percent difference in the answer, so it is easier to make decisions about acceptable error.
`\qquad\qquad\qquad "lg"(72900) \approx 4_{20}17_{12}3;" pt"`,
`\qquad\qquad\qquad = (80 + 17)_{12}3;" pt"`,
`\qquad\qquad\qquad = 97 xx 12 + 3" pt"`,
`\qquad\qquad\qquad = 1164 + 3" pt"`,
`\qquad\qquad\qquad = 1167" pt"`.
Suppose we wanted to find `72900^{3//7}`.
The logarithm of `72900^{3//7}` is `3//7` of the logarithm of `72900`, i.e. about `3//7` of `1167` points. `1167 xx 3 = 3501`. `3501 -: 7 \approx 500.1`. So the logarithm of our answer is aboout `500.1" pt"`.
Reverting this into mixed units, we use divmod division, first by 12, and then by 20.
`\qquad\qquad\qquad 500.1 -: 12 = 41;_{12}8.1`, and
`\qquad\qquad\qquad 41 -: 20 = 2;_{20}1`,
so that
`\qquad\qquad\qquad 500.1 = 2_{20}1_{12}8.1`,
and
`\qquad\qquad\qquad "lg"(72900^{3/7}) \approx 2" dec" + 1" dB" + 8.1" pt"`.
Then
`\qquad\qquad\qquad 72900^{3/7} \approx 10^2 xx 1.12 xx 1.081`,
`\qquad\qquad\qquad\qquad = 121.`,
which compares well with the more exact value `121.346`, especially considering that table values such as `1.12` have already been rounded to three significant figures.
