Математика / 4.Прикладная математика.

Glebov V.V., Danilova T.G., Rustrepenin D. A.

The College of Railway Transport,

The Ural State University of Railway Transport,

The city of Yekaterinburg.

Calculation of logarithms with any base

Every high school student is able to calculate any logarithm by means of the calculator, but even not every teacher of mathematics is in power to find a value of any logarithm on the sheet of paper.

150 years ago Professor Sarrus from Strasbourg offered a very easy way of calculation of common logarithms. Using a formula of transition to the new base, any logarithm can be represented by a common logarithm and consequently, this method will be suitable for calculation of any logarithms [1].

Calculations by Sarrus's method are made in a binary numeral system, therefore at first we will address to the habitual decimal system, and we will see how to pass to binary one .

In the commonly accepted decimal numeral system each number is considered as the sum of various degrees of number 10 which is the base of the decimal system. For example:

According to such notation of the number only coefficients of number 10 in various degrees are written out. As all these coefficients don't exceed ten, for their notation only one of ten digits suffices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The position taken by the digit in the writing of the number is noted by the power of ten coefficient of which is the considered digit. Counting of serial positions in the number is conducted from 0 and begins from a decimal point. The first digit to the left of a decimal point corresponds to zero degree of number 10 (the category of ones), the second digit before a decimal point means the first degree of number 10 (the category of tens), the first digit on the right of a decimal point corresponds the minus first degree of number 10 (the category of the tenth), etc.

In a binary numeral system the base is number 2, and all numbers in this system are considered as the sum of various degrees of number two. For example, number 21.25 is written as follows:

To represent a number in the binary numeral system, it is necessary to write down coefficients of number 2 in various degrees consistently. The decimal point should be written after the coefficient corresponding to the two in zero degree. So, 21.25 in a binary numeral system is 10101.01. Coefficients of number 2 in various degrees can have only two values: 1 or 0. Therefore, only two of these digits are necessary for the notation of a number in the binary system. To multiply a number in the binary numeral system by 2, it is necessary to move a decimal point one digit to the right. This action is the basis of Sarrus' method. [2]

Let , then x =lgA, where lgA is a common logarithm of number A. The task is to find X at given A. The characteristic of a logarithm (i.e. its whole part) can be found easily. Let it be equal р, then we will divide both parts of the initial equation by 10 in the p degree: , and now we note (x-p) for y, and for B, then the equation will take a form: , at the same time the characteristic of number y is equal to 0. Now it is necessary to find a number mantissa (a fractional part of a logarithm of number).

Let's assume that the number y is represented in the binary system:

, where are consecutive digits of one number (i.e. units and zeroes):

Further we square the resulting equation. For this purpose it is necessary to double an exponent in the left part of the equation (indicators are multiplied at raising degree to degree), i.e. to move a decimal point one digit to the right.

We have: , is the characteristic of a logarithm of number and is equal to 1 if and is equal to 0, if . Now, having found the digit , we divide the last equality by , and we get: = . By analogy we find the following digit - .. So, It is possible to find any number of digits of the logarithm which is written down in the binary numeral system by such a simple way.

Literature:

1. Algebra and introduction to calculus 10-11. Sh. Alimov, 254 pp.

2. The planet of informatics. S. Shaposhnikova.