The Middle Ages are known as travel times and geographical discoveries. The only way to implement long trips was navigation, which is always associated with the implementation of large volumes of navigation calculations. Now it is difficult to imagine the process of exhausting calculations when multiplying-dividing five or six-digit numbers “manually”. John Napier, a theologian by the nature of his main activity, engaged in trigonometric calculations at his leisure, guessed to replace the laborious procedure of multiplication with simple addition. He himself said that his goal was "to free oneself from the difficulties and boredom of computing, which scare many away from the study of mathematics." The efforts were successful - a mathematical apparatus was created, called the system of logarithms.
So what is the logarithm? The basis of the logarithmic calculations is a different representation of the number: instead of the usual positional system, as we are used to, the number A is represented as a power expression, where some arbitrary number N, called the base of the degree, is raised to such a degree n that the result is the number A. Thus , n is the logarithm of the number A on the basis of N. The choice of the base of the logarithms determines the name of the system. For simple additions, the decimal system of logarithms is used, and in science and technology the system of natural logarithms is widely used, where the basis is the irrational number e = 2.718. The expression that defines the logarithm of the number A is written in the language of mathematics as follows:
n = log (N) A, where N is the base of the degree.
Decimal and natural logarithms have their own specific abbreviated spelling - lgA and lnA, respectively.
In a calculation system using the calculation of logarithms, the main element is the conversion of the number to a power law using a table of logarithms for some reason, for example 10. This manipulation does not present any difficulties. Further, the property of power numbers is used, which consists in the fact that when multiplied, their degrees add up. In practice, this means that the multiplication of numbers with a logarithmic representation is replaced by the addition of their degrees. Therefore, the question “what is the logarithm”, if we continue it to “why do we need it”, has a simple answer - to simplify the procedure of multiplication-division of multi-digit numbers - after all, adding “in a column” is much easier than multiplying “in a column”. Who does not believe - let him try to add and multiply two eight-digit numbers.
The first tables of logarithms (based on a natural number) were published by John Napier in 1614, and a completely error-free version, including tables of decimal logarithms, appeared in 1857 and is known as Bremiker's tables. The use of logarithms with a base in the form of an irrational number is due to the fact that the number e is quite simple to obtain through the Taylor series, which is widely used in integral and differential calculus.
The essence of this computing system is contained in the answer to the question “what is the logarithm” and follows from the main logarithmic identity: N (the base of the logarithm) raised to the power n, equal to the logarithm of the number A (logA), is equal to this number A. Moreover, A> 0, those. the logarithm is determined only for positive numbers, and the base of the logarithm is always greater than 0 and not equal to 1. Based on the foregoing, the properties of the natural logarithm can be formulated as follows:
- The domain of the natural logarithm is the entire numerical axis from 0 to infinity.
- ln x = 0 - a consequence of the known relation - any number in the zero degree is 1.
- ln (X * Y) = ln X + lnY - the most important property for computational manipulations is the logarithm of the product of two ramen numbers, the sum of the logarithms of each of them.
- ln (X / Y) = ln X - lnY - the logarithm of the quotient of two numbers is equal to the difference of the logarithms of these numbers.
- ln (X) n = n * ln X.
- The natural logarithm is a differentiable, convex upward function, with ln 'X = 1 / X
- log (N) A = K * ln A - the logarithm for any positive and non-e basis is different from the natural one only by a coefficient.
Now every student knows what the logarithm is, but thanks to progress in the field of applied computer technology, the problems of computing work are a thing of the past. Nevertheless, logarithms, already as a mathematical tool, are used in solving equations with unknowns in the exponent, in expressions for finding the decay time of radioactive elements and in other areas of mathematics, physics, and statistics.