the Middle Ages known as the time of travel and geographical discoveries.The only way to realize long-distance travel was sailing that is always associated with the implementation of large amounts of navigational calculations.It is difficult to imagine the grueling process of calculations for multiplication, division of five-digit numbers "by hand".John Napier, a theologian by the nature of its core business, being engaged at leisure trigonometric calculations guessed replace laborious process of multiplication by simple addition.He said that his goal was to "get rid of difficulties and boredom calculations that deters many from the study of mathematics."The efforts were crowned with success - was created mathematical apparatus, called the system of logarithms.
So, what is a logarithm?The basis of calculation is different logarithmic representation of the number: instead of the usual positional system as we used, the number of A is represented in the form of a power of expression, which some arbitrary number N, called a base point is raised to such a degree n, which is resulting in a number of A. Thus, n - is the logarithm of A to base N. Selecting the base of logarithms determines the name of the system.For simple vichisleny used the decimal system of logarithms, and in science and technology widely used system of natural logarithms, wherein the base is an irrational number e = 2.718.The expression defining the logarithm of the number of A, the language of mathematics is written as:
n = log (N) A, where N - the foundation degree.
Decimal and natural logarithms have their specific abbreviated writing - lgA and lnA, respectively.
The payment system that uses the calculation of logarithms, the main element is the transformation of mind to the power using a table of logarithms of some base, for example 10. This manipulation does not present any difficulties.Then use the property power of numbers, consisting in the fact that when multiplied by the degree of their fold.Practically, this means that the multiplication of logarithmic representation numbers, is replaced with addition of their degrees.Therefore, the question "what is the logarithm of the" if it is to continue to "and why we need it," is the simple answer - to simplify the procedure for the multiplication-division multiple-bit numbers - after the addition of "in a column" is much easier to multiply "in a column."Who does not believe - let him try to lay down and to multiply two eight-bit numbers.
first tables of logarithms (to the base with a natural number), published in 1614, John Napier, and completely free of errors option, and includes a table of common logarithms, appeared in 1857 and is known as a table Bremikera.Using the logarithm to the base in the form of an irrational number because the number is, quite simply to get through the Taylor series, which has wide application in integral and differential calculus.
essence of this computer system is contained in the answer to the question "what is the logarithm of the" in and out of the main logarithmic identity: N (base of logarithm) raised to the power n, equal to the logarithm of the number of A (logA), equal to the number of A. Moreover, A & gt;0;logarithm is defined only for positive numbers, and the base of the logarithm is always greater than 0 and equal to 1. Based on the foregoing, the properties of the natural logarithm can be summarized as follows:
- domain of the natural logarithm - the whole real axis from 0 to infinity.
- ln x = 0 - a consequence of the known relation - any number to the zero power is equal to 1.
- ln (X * Y) = ln X + lnY - most important for computer manipulation of the property - the logarithm of the product of two numbers ramen sum of the logarithms of each of them.
- ln (X / Y) = ln X - lnY - logarithm of the quotient of two numbers is equal to the difference between the logarithms of those numbers.
- ln (X) n = n * ln X.
- natural logarithm is a differentiable convex upward function and ln 'X = 1 / X
- log (N) A = K * ln A - logarithm of any positive and different from the number of e base differs from the natural only factor.
now every schoolchild knows that such a log, but thanks to advances in the field of applied computing problems of computational work are gone.However, logarithms, already as a mathematical tool used in solving equations with unknowns in the exponent, in terms of time to find the decay of radioactive elements in other areas of mathematics, physics and statistics.
