Logarithms

Logarithms, a series of numbers having a certain relation to the series of natural numbers, by means of which many arithmetical operations are made comparatively easy. The nature of the relation will be understood by considering two simple series such as the following, one proceeding from unity in geometrical progression, the other from 0 in arithmetical progression:

Geom. series, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, &c.
Arith. series, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, &c.

Here the ratio of the geometrical series is 2, and any term in the arithmetical series expresses how often 2 has been multiplied into 1 to produce the corresponding term of the geometrical series; thus, in proceeding from 1 to 32, there have been 5 steps or multiplications by the ratio 2; in other words, the ratio of 32 to 1 is compounded five times of the ratio of 2 to 1. It was this conception of the relation that led to giving the name of Logarithms to the terms of the arithmetical series, the word logarithm (Gr. logōn arithmos) meaning 'the number of the ratios.' As to the use that may be made of such series, it will be observed that the sum of any two logarithms (as we shall now call the terms of the lower series) is the logarithm of their product—e.g. 9 (= 3 + 6) is the logarithm of 512 (= 8 × 64). Similarly, the difference of any two logarithms is the logarithm of the quotient of the numbers; a multiple of any logarithm is the logarithm of the corresponding number raised to the power of the multiple—e.g. 8 (= 4 × 2) is the logarithm of 256 (= 16²), and a submultiple of a logarithm is the logarithm of the corresponding root of its number. In this way, with complete tables of numbers, and their corresponding logarithms, addition is made to take the place of multiplication, subtraction of division, multiplication of involution, and division of evolution.

In order to make the series above given of practical use, it would be necessary to complete them by interpolating a set of means between the several terms, as will be explained below. We have chosen 2 as the fundamental ratio, or base, as being most convenient for illustration; but any other number (integral or fractional) might be taken; and every different base, or radix, gives a different system of logarithms. The system now in use has 10 for its base; in other words, 10 is the number whose logarithm is 1.

The idea of making use of series in this way would seem to have been known to Archimedes and Euclid, without, however, resulting in any practical scheme; but by the end of the 16th century trigonometrical operations had become so complicated that the wits of several mathematicians were at work to devise means of shortening them. The real invention of logarithms is now universally ascribed to John Napier (q.v.), Baron of Merchiston, who in 1614 printed his Canon Mirabilis Logarithmorum. His tables only give logarithms of sines, cosines, and the other functions of angles; they also labour under the three defects of being sometimes + and sometimes -, of decreasing as the corresponding natural numbers increase, and of having for their radix (the number of which the logarithm is 1) the number which is the sum of $1\; +\; 1\; +\; \backslash frac\{1\}\{1.2\}\; +\; \backslash frac\{1\}\{1.2.3\}\; +$, &c. In many calculations, however, the latter is an advantage rather than a defect. These defects were, however, soon remedied: John Speidell in 1619 amended the tables in such a manner that the logarithms became all positive, and increased along with their corresponding natural numbers. He also, in the sixth edition of his work (1624), constructed a table of Napier's logarithms for the integer numbers, 1, 2, 3, &c., up to 1000, with their differences and arithmetical complements, besides other improvements. Speidell's tables are now known as hyperbolic logarithms. But the greatest improvement was made in 1615, by Professor Henry Briggs (q.v.), of London, who substituted for Napier's inconvenient 'radix' the number 10, and succeeded before his death in calculating the logarithms of 30,000 natural numbers to the new radix. Briggs's exertions were ably seconded; and before 1628 the logarithms of all the natural numbers up to 100,000 had been computed. Computers have since chiefly occupied themselves rather in repeatedly revising the tables already calculated than in extending them.

Construction of Tables.—The following is the simplest method of constructing a table of logarithms on Briggs's system. The log. of 10 = 1; the log. of 100 (which is twice compounded of 10) = 2; the log. of 1000 = 3, &c.; and the logarithms of all powers of 10 can be found in the same manner. The intermediate logarithms are found by continually computing geometric means between two numbers, one greater and the other less than the number required. Thus, to find the log. of 5, take the geometric mean between 1 and 10, or 3.162..., the corresponding arithmetic mean (the log. of 1 being 0, and that of 10 being 1) being 0.5; the geometric mean between 3.162... and 10, or 5.623..., corresponds to the arithmetic mean between 0.5 and 1 or 0.75; the geometric mean between 3.162... and 5.623..., or 4.216..., has its logarithm = $\backslash frac\{1\}\{2\}$(0.75 + 5) or 0.625; this operation is continued till the result is obtained to the necessary degree of accuracy. In this example the twenty-first result gives the geometric mean = 5.000,003, and the corresponding arithmetic mean = 0.698,970, which is in ordinary calculations used as the logarithm of 5. Since division of numbers corresponds to subtraction of logarithms, and since $2\; =\; \backslash frac\{10\}\{5\}$, the log. of 2 = log. 10 - log. 5 = $1\; -\; 0.698970\; =\; 0.301030$. The logarithms of all prime numbers are found in the same way as that of 5; those of composite numbers are obtained by the addition of the logarithms of their factors; thus the log. of 6 = log. 2 + log. 3 = 0.301030 + 0.477121 = 0.778151. This method, though simple in principle, involves an enormous amount of calculation; and the following method, which depends on the modern algebraic analysis, is much to be preferred. According to this method, logarithms are considered as indices or powers of the radix; thus, $10^0\; =\; 1$, $10^\{0.301030\}\; =\; 2$, $10^\{0.477121\}\; =\; 3$, $10^2\; =\; 100$, &c.; and the laws of logarithms then become the same as those of indices. Let $r$ represent the radix, $y$ the natural number, $x$ its logarithm; then $y\; =\; r^x$, or, putting $1\; +\; a$ for $r$, $y\; =\; (1\; +\; a)^x$; and it is shown by the binomial and exponential theorems (see the ordinary works on Algebra) that $y\; =\; 1\; +$ , where , the former equation expressing a number as the sum of different multiples of its logarithm and the radix. If $1/p$ be substituted for $x$, then $y\; =\; 1\; +\; 1\; +\; \backslash frac\{1\}\{1.2\}\; +\; \backslash frac\{1\}\{1.2.3\}\; +$ which, as before mentioned, is Napier's radix, and is generally called $e$. Hence $r\; =\; e^p$, or $p$ is the logarithm of $r$ to the base or radix $e$. Then, referring to the above-mentioned value of $p$, we have $\backslash log\_e\; r$ (i.e. log. of $r$ to the base $e$) , or, as before, putting $1\; +\; a$ for $r$, ; a series from which $\backslash log\_e\; (1\; +\; a)$ cannot be found, unless $a$ be a proper fraction. But if we put $-a$ for $a$, ; and, subtracting this expression from the former, $\backslash log\_e\; (1\; +\; a)\; -\; \backslash log\_e\; (1\; -\; a)$ or ; and, for the sake of convenience, putting $(u\; +\; 1)/u$ for $(1\; +\; a)/(1\; -\; a)$, in which case $a\; =\; 1/(2u\; +\; 1)$, we finally obtain , or . If 1 be put for $u$ in this formula, the Napierian logarithm of 2 is at once obtained to any degree of accuracy required; if 2 be put for $u$, the Napierian logarithm of 3 can be calculated, &c. Now, as logarithms of any system have always the same ratio to one another as the corresponding logarithms of any other system, no matter what its base, if a number can be found, which, when multiplied into the logarithm of a certain number to one base, gives the logarithm of the same number to another base, this multiplier will, when multiplied into any logarithm to the first base, produce the corresponding logarithm to the other base. The multiplier is called the modulus, and, for the conversion of Napierian into common or Briggs's logarithms, is equal to 0.4342944...; so that, to find the common logarithm of any number, first find the Napierian logarithm, and multiply it by 0.4342944...

As in Briggs's system the logarithm of 10 is 1, and that of 100 is 2, it follows that all numbers between 10 and 100 have for their logarithms unity + a proper fraction; in other words, the integer portion of the logarithms of all numbers of two figures is unity; similarly, the integer portion of the logarithms of numbers between 100 and 1000 is 2, and, in general, the integer portion of the logarithm of any number expresses a number less by unity than the number of figures in that number. This integer is called the characteristic, the decimal portion being the mantissa.

As the logarithm of 1 = 0, the logarithms of quantities less than unity would naturally be negative; thus, the logarithm of $\backslash frac\{1\}\{2\}$ would be $-0.30103$. But, for convenience in working, the mantissa is kept always positive, and the negative sign only affects the characteristic; the logarithm of $\backslash frac\{1\}\{2\}$ or 0.5 would thus be $\backslash bar\{1\}.69897$, the characteristic in this and similar cases expressing, when the fraction is reduced to a decimal, the number of places the first figure is removed from the decimal point; thus, the logarithm of 0.0005 is $\backslash bar\{4\}.69897$.

Directions for the use of logarithms in calculation will be found prefixed to any set of mathematical tables.

The tables most distinguished for accuracy are the French ones of Callet, Lalande, Bagays; Hutton's, those which Babbage produced with the aid of his calculating machine, Shortrede's, and Sang's; and the German ones of Gauss, Schrön, Bruhns, Von Vega, Bremiker. A serviceable handbook is Chambers's Mathematical Tables, edited by Pryde.