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Log rules and a historical background of logarithm application

‘Logarithm’ is a term coined by Napier, which is derived from the Greek words logoz and ariumoz – the combination of these two words literally means ‘numbers of relations’. Therefore, one may rightfully say that logarithms and log rules were invented and first put into a practical use by Napier. The need for complex calculations in the XVI century grew rapidly, and a considerable amount of difficulties was associated with multiplication and division of multi-digit numbers. At the end of the century several mathematicians, almost simultaneously, came up with an idea to replace a time-consuming multiplication by simple addition, and in this case, division can be automatically replaced by an infinitely more simple and reliable subtraction; finally, extraction of the root of degree n can be reduced to division of the logarithm of a radicand by n. For the first time, this idea was published in the book «Arithmetica integra» by Michael Stifel, who, however, did not put a serious effort in implementing the mathematical insights of his.

In 1614, the Scottish mathematician John Napier Amateur published the essay in Latin, which concerned an amazing nature of logarithms and included a comprehensive discussion over this mathematical conception. Primarily, it featured a brief description of logarithms and their properties, as well as the 8-digit tables of logarithms of sines, cosines and tangents, with a step of 1'. That was the time when the term logarithm proposed by Napier became well established in science and began perturbing scientists’ minds. The concept of functions did not yet exist, and Napier determined his understanding of logarithm cinematically by comparing uniform and logarithmically-slow motion. Napier formulated the main characteristic of a logarithm in the following way: if values form a geometric progression, then their logarithms form an arithmetic progression. However, the log rules for the Napier function are different from the rules for modern logarithm. Unfortunately, all the values of the Napier table contained processing error after the sixth sign. No sooner than in 5 years, in 1619, the London math teacher John Speidell reissued Napier table converting them in such a way that they actually became tables of natural logarithms (though Spaydell retained the scaling to integers). The term natural logarithm was suggested by the Italian mathematician Pietro Mengoli in the middle of the XVI century.

The concept of logarithms and log rules that were closer to the modern understanding of the issue (as an operation reverse to raising to power) first appeared in the works by Wallis and Johann Bernoulli, and Euler finally legalized the use of the concept in the XVIII century. In his book related to an extended study of some infinite mathematical notions (1748), Euler gave a modern definition of the exponential and logarithmic functions as well as log rules, provided the expansion of their power series as well as highlighted the role of the natural logarithm. Finally, in the 1620s, Edmund Wingate and William Oughtred invented a slide rule, and it used to be an invaluable tool before the advent of pocket calculators.

Log rules, a logarithmic function and its graph

If we consider the index x in the equation y = ax as an independent variable, then y becomes a function of x, and the latter is called an exponential function. But if we consider y in this equation as an independent variable, then x will be a function of y, i.e. x is the logarithm of a number in the base and that can be expressed as: x = loga y. Designating an independent variable as x and indicating the function of this variable with the letter y (i.e. replacing x by y, and vice versa), we can have the same function written as: y = loga x. The latter function is called a logarithmic function. Having log rules and the graph of a logarithmic function on hand, we can find an approximate logarithm of a number, which is placed between the maximum and minimum taken values of x.

When considering the drafted graphs, we can clearly understand the following properties of logarithms and log rules:

  • 1) As the charts are entirely located to the right of the y-axis, therefore it is evident that negative numbers do not have logarithms (remember that for any value of x, the function ax is positive)
  • 2) it is well known that a certain ordinate always corresponds to a positive abscissa; therefore, any positive number has a logarithm
  • 3) All the curves intersect with the x-axis at the same point that is distant from the origin by + l. This means that for each radix the logarithm of 1 is zero (a0 = 1)
  • 4) When a> 1, then the parts of the curves corresponding to abscissae less than 1 lie in the angle x0y', whereas the parts of the curves corresponding to abscissae greater than 1 are located at the angle x0y. This means that if the radix is greater than 1, logarithms of numbers less than 1 are negative, and logarithms of numbers greater than 1 are positive. This fully corresponds to log rules and the property of the exponential function according to which positive values of x cause the function ax to be greater than 1, while for the negative ones the function is less than 1 (if a> 1). Apparently, for a <1 (for example, for the curve y = log ½ x) the conclusion is opposite
  • 5) The logarithm of the radix itself is equal to 1. Thus, it can be seen on the graph y = log2 x that the abscissa 2 corresponds to the ordinate 1; the same thing takes place on the other graphs.
  • 6) In case of radix being greater than 1 the branches of the curves located below the axis x close to the semi-axis 0y' with the decreasing of abscissas from 1 to 0; however, the branches never reach it, while the branches of the same curve disposed above the axis x rise higher and higher infinitely with x increasing from 1 to +∞. The latter means that (for a> 1), with numbers increasing from 0 to 1, the logarithm increases from -∞ to 0; with an increase of a number from 1 to +∞ the logarithm of it increases from 1 to +∞. Consequently, one knowing log rules can conclude that a larger number corresponds to a larger logarithm (if its radix is less than 1 (a <1), the conclusion will be opposite).

Log rules: finding the logarithm of a product and properties of logarithms

Let us suppose we want to perform the following multiplication: 378 * 45.2. Let us try to perform this operation by means of logarithms. At first, we need to find the logarithms of the numbers 378 and 45.2 in the tables. For instance, those logarithms are 2.5775 and 1.6551 (at radix 10). It means that 378 = 102,5775 and 45,2 = 101,6551; therefore, 378 * 45,2 = 102,5775 * 101,6551. As long as the multiplication of the powers of the same number implies the addition of the indices of these powers (whatever are the indices), we know that 378 * 45.2 = 102.5775 + 1.6551 = 104.2326. Therefore, the logarithm of the product 378 * 45.2 is the number 4.2326 resulting from the addition of the logarithms of these factors (we can find the product itself using the logarithm tables).

Supposing that N1 and N2 are two numbers whose product is to be calculated we have to find the logarithms of these numbers in the tables (x1 and x2). The radix of the logarithms can be 10, as well as any other number, which we denote a. Taking into consideration log rules we have the equations: N1 = ax1; N2 = ax2 and, consequently, N1N2 = ax1 * ax2 = ax1+x2. This shows us that log (N1N2) = x1 + x2. However, x1 is log N1, and x2 is log N2 which means: log (N1N2) = log N1 + log N2, i.e. the logarithm of the product (at whatever radix) is equal to the sum of the logarithms of the factors (taken at the same radix). In conclusion, this remains true even when there are more factors than 2, as in case of the multiplication of the powers of the same number their indexes are added, and this is also true when there are more powers than 2.

In order to understand log rules better let us take an integer that is not depicted as 1 with zeros, for example, 35, or an integer and a fraction, for example, 10.7. The logarithm of such a number cannot be an integer, since as we raise 10 to a power with a whole number as an index (positive or negative), we obtain 1 with zeros (following 1 or preceding it). We now can assume that the logarithm of this number is any fraction of the kind a/b. Resting on the aforementioned properties of the logarithmic function we know that every positive number has a logarithm; therefore, each of the numbers 35 and 10.7 have their logarithm, and since they cannot be whole numbers or fractional numbers, there are all grounds to stick to the assumption that the corresponding numbers are irrational, and therefore they cannot be expressed exactly by ciphers. Usually, irrational logarithms are approximately expressed as decimal fractions with a few decimal characters. An integer of this fraction (even if it is as much as 0) is called ‘characteristic’, and the fractional part is known as the mantissa of a logarithm. If, for example, there is a logarithm 1.5441, the characteristic of it is 1 and the mantissa amounts to 0.5441. Thus, the characteristic of the logarithm of a decimal less than 1 contains as many negative units as there are zeros in the decimal before the first significant figure including zero integers; the mantissa of the logarithm is positive.

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