48.

The main advantage of the use of logarithms is that multiplication and division are reduced to addition and subtraction, i.e., to a lower calculation mode. However, since addition and subtraction are already the lowest modes, there exist no rules for the computation of log (a + b) or log (a - b). The advantages of logarithms begin therefore with multiplication.

	N	log		N	log
a = 1.32	a	0².1206^-2	a = 0.00931	a	0.9689 -3
b = 24.1	b	1.3820 [-	b = 0.0000148	b	0.1703 -5 [-
a/b= 0.0548	a/b	0.7386 -2	a/b = 629	a/b	2.7986

The lay-out of the preceding table suits the evaluation of logarithms. If you had formed the difference log a - log b directly, you would have obtained a negative decimal fraction, that is, it would not be in the form 0.··· -n. In order to obtain this immediately, you place in log a instead of the characteristic 0 the characteristic 2,-2. Note that in the second example, as you subtract the difference 0.1703 -5, the number becomes +5.

	N	log		N	log
a = 0.846	a	0.9274 - 1 [.8	a = 0.846	a	.9274 [:5
	a⁸	7.4192 - 8
a⁸= 0.263	a⁸	0.4192 - 1	= 0.967		0.9855 - 1

In the second example, the division (0.9274 -1):5 would yield 0.1855 - 1/5, and you would have to add and subtract 0.8, in order to find the characteristic in the proper form. In order to avoid this, you write log a = 4.9274 - 5 and only then divide by 5, in order to obtain the characteristic in the form 0.··· -1.

We only know logarithms of positive numbers, whence we start with

and take account of the minus sign at the end of the calculation. You must compute separately a³ and b³ .

	N	log	N	log
a = 0.814	a	0.9106 - 1 [.3	s	0.3636
a³ = 0.539	a³	2.7318 - 3	a	0.9106 - 1
		0. -1	b	0.0828
b = 1.21	b	-.0828 [-3	s/ab	0.3702 [:3
b³ = 1.77	b³	0.2484
a³ + b³ = 2.309 = s			=1.33	=0.1234

In the table, log 20.3 = 1.3075 and log 20.4 = 1.3096. When you wanted to obtain log 20.36, you have rounded the number to three digits. Obviously, log 20.36 will lie between the two values given in the table. In order to find a more accurate value of the logarithm, you assume that it changes linearly within the table interval under consideration; you can do so provided the interval is sufficiently small compared with the desired accuracy. This means, in geometrical terms, that you replace the logarithmic curve by a segment of a straight line. Thus, the intermediate values between two values in the table are determined by proportions:

log20.30=1.3075		If the argument 20.30 rises to 20.40 by 10 units, the mantissa rises from 3075
		to 3096 or by 21 units. If the argument rises by 1 unit in the last place, then the
log 20.36 = ····		mantissa rises by by 21/10 units in the last place. Therefore, if the argument
		rises by 6 units in the last place, the mantissa increases by 21·6/10 or 126/10
log 2..40=1.3096		units in the last place.

Since the mantissae in a four-place table are rounded off,
it is useless to exceed this number of digits!

The same result would have been obtained directly by a proportion. For example, let 10 units of the last digit of the argument be related to 6 units like 21 units of the last digit in the mantissa to x units:

Denote the table difference, the number 21 above, by D and the number 6 by n, then, in general, x = D·n/10, a formulae, from which you can obtain x directly.

In order to save effort, a proportion table is attached to logarithmic tables; a section is shown here. The quantity n, which is a part of the argument, is listed in the vertical column on the left hand side, the mantissae difference D in the first line at the top. You find the increase in the mantissa in the table, corresponding to given values of n and D. In the example above, D = 21, n = 16. The table shown yields the increase in the mantissae of 13, in agreement with the earlier calculation.

If you want to find n from the proportion table, you find for D = 24 the increase in the mantissa 14 and in the left column n = 6.

The determination of intermediate values is called interpolation; if you assume linear growth, it is called linear interpolation.

Since the last digit of the mantissa is rounded, in a certain sense, a computation will always be inaccurate. However, as a rule, data are inaccurate, because most frequently they have been measured.

For example, if you measure the three angles of a triangle and do not cheat, they will normally not add up to 180º, if you extract values from a curve, you will not achieve complete accuracy, etc. It is a psychological fact that two points can be distinguished from each other, if they have a certain distance in between. If this distance is too small, you will see only one point, although there are really two points. Direct measurement is no longer possible and an optical tool must be employed. And even that has its limitations. Its enlargement is subject to diffraction, which again limits accuracy. If you use yet finer methods, there will always arise limiting bounds, which cannot be overcome. Hence any data which exceed these limitations are senseless and either rest on ignorance or deceit. In some advertisements, for example, those concerning the salinity of fountains in spas, you will see numbers after the decimal point which can hardly be determined, apart from the fact that the mineral content is not constant. In practice, as a rule, you measure not even to the highest level of science, but with an accuracy which you demand for practical reasons. For example, a distance a will be measured so that it lies between 1.21 and 1.22 cm. If you now compute a² and a³ for the two bounds and their mean value:

a = 1.21	a = 1.215	a = 1.22
a² = 1.4641	a² = 1.476225	a² = 1.4884
a³ = 1.771561	a³ = 1.793613375	a³ = 1.815848

In practice, frequently three or four digits are valid, whence you need only four-digit logarithm tables. This does not mean, that they will always be sufficient. For example, for calculations of the stock market, interest of interest calculations and such like, you often have to deal with more than four digits and then look for tables with larger digits.

It is of interest to know that Briggs used logarithms with 14 decimals for very accurate calculations. His tables were later completed and rounded off to 10 decimals. John Couch Adams (1819 - 1892) achieved for the numbers 1 - 109 the largest number of decimals with 260 places. Tables with different numbers of decimals have been published (48, 20, 15, ···). They are so huge that they are not easily carried about; moreover the copying out of so many digits is very time consuming. Only recently, the emphasis on this number luxury has changed and has been shifted to the right relationship between the number of digits and attainable accuracy. For a long time, students in schools calculated with 7-digit logarithms. At the end of the 19th Century, their tables contained 5-digits logarithms until in the 1930s the tables were replaced by 4-digits tables.

For certain purposes, 3-digits logarithms are sufficient and one can also use the slide rule. Of course, electronic calculators have displaced all these means of calculation and found use by a much wider public.

For a complete calculation, you must state the accuracy of 1. the input and 2. the output. We will not treat this subject here in great detail and only demonstrate by means of an example, how you can determine from exact, given data the largest possible error during a logarithmic computation.

Errors can be positive and negative and can cancel each other during a computation, but they can also add up. The largest possible error during addition and subtraction of logarithms is the sum of the errors. The error grows during multiplication, decreases during division.

N	log	max.error	The table yields log 6.25 = 0.7959. As it has
a	0.7959	0.00005	been rounded, the value can have arisen from
b	0.6848	0.00005	a value between 0.79585000 and 0.795149999.
c	0.7589	0.00005	The largest error in the 4th decimal is 0.00005.
			The same observation applies to the other
abc	2.2396	0.00015	logarithms.
d	0.9112	0.00005
			The value of log x must lie between
abc/d	1.3284	0.0002	0.6641 and 0.6443, so that the four digit
x	0.6642	0.0001	rounded argument is x = 4.617 or x = 4.614.

log 18.20		1.2601	a = 18.20	a = 18.26
	D = 24
log 18.20		1.2625	n = 6	18.20 + 6 = 18.26

log a³= 3log a		log b³= 3log b		= (log s - log a - log b)/3