Introduction
Logarithms were invented to speed up and simplify calculations. The idea of a logarithm, that is, the idea of expressing numbers as powers of the same base, belongs to Mikhail Stiefel. But in Stiefel’s time, mathematics was not so developed and the idea of the logarithm was not developed. Logarithms were later invented simultaneously and independently of each other by the Scottish scientist John Napier (1550-1617) and the Swiss Jobst Burgi (1552-1632). Napier was the first to publish the work in 1614. entitled "Description of the amazing table of logarithms", Napier's theory of logarithms was given in sufficient in full, the method for calculating logarithms is given the simplest, therefore Napier’s merits in the invention of logarithms are greater than those of Bürgi. Burgi worked on the tables at the same time as Napier, but kept them secret for a long time and published them only in 1620. Napier mastered the idea of the logarithm around 1594. although the tables were published 20 years later. At first he called his logarithms “artificial numbers” and only then proposed to call these “artificial numbers” in one word “logarithm”, which translated from Greek means “correlated numbers”, taken one from an arithmetic progression, and the other from a geometric progression specially selected for it. progress. The first tables in Russian were published in 1703. with the participation of a wonderful teacher of the 18th century. L. F. Magnitsky. In the development of the theory of logarithms great importance had works by St. Petersburg academician Leonhard Euler. He was the first to consider logarithms as the inverse of raising to a power; he introduced the terms “logarithm base” and “mantissa.” Briggs compiled tables of logarithms with base 10. Decimal tables are more convenient for practical use, their theory is simpler than that of Napier’s logarithms . Therefore, decimal logarithms are sometimes called Briggs logarithms. The term "characterization" was introduced by Briggs.
In those distant times, when the sages first began to think about equalities containing unknown quantities, there were probably no coins or wallets. But there were heaps, as well as pots and baskets, which were perfect for the role of storage caches that could hold an unknown number of items. In the ancients mathematical problems Mesopotamia, India, China, Greece, unknown quantities expressed the number of peacocks in the garden, the number of bulls in the herd, the totality of things taken into account when dividing property. Scribes, officials and priests initiated into secret knowledge, well trained in the science of accounts, coped with such tasks quite successfully.
Sources that have reached us indicate that ancient scientists had some general techniques for solving problems with unknown quantities. However, not a single papyrus or clay tablet contains a description of these techniques. The authors only occasionally provided their numerical calculations with skimpy comments such as: “Look!”, “Do this!”, “You found the right one.” In this sense, the exception is the “Arithmetic” of the Greek mathematician Diophantus of Alexandria (III century) - a collection of problems for composing equations with a systematic presentation of their solutions.
However, the first manual for solving problems that became widely known was the work of the Baghdad scientist of the 9th century. Muhammad bin Musa al-Khwarizmi. The word "al-jabr" from the Arabic name of this treatise - "Kitab al-jaber wal-mukabala" ("Book of restoration and opposition") - over time turned into the well-known word "algebra", and al-Khwarizmi's work itself served the starting point in the development of the science of solving equations.
Logarithmic equations and inequalities
1. Logarithmic equations
An equation containing an unknown under the logarithm sign or at its base is called logarithmic equation.
The simplest logarithmic equation is an equation of the form
log a x = b . (1)
Statement 1. If a > 0, a≠ 1, equation (1) for any real b has a unique solution x = a b .
Example 1. Solve the equations:
a)log 2 x= 3, b) log 3 x= -1, c)
Solution. Using Statement 1, we obtain a) x= 2 3 or x= 8; b) x= 3 -1 or x= 1 / 3 ; c)
or x = 1.Let us present the basic properties of the logarithm.
P1. Basic logarithmic identity:
Where a > 0, a≠ 1 and b > 0.
P2. Logarithm of the product of positive factors equal to the sum logarithms of these factors:
log a N 1 · N 2 = log a N 1 + log a N 2 (a > 0, a ≠ 1, N 1 > 0, N 2 > 0).
Comment. If N 1 · N 2 > 0, then property P2 takes the form
log a N 1 · N 2 = log a |N 1 | + log a |N 2 | (a > 0, a ≠ 1, N 1 · N 2 > 0).
P3. The logarithm of the quotient of two positive numbers is equal to the difference between the logarithms of the dividend and the divisor
(a
> 0, a
≠ 1, N
1
> 0, N
2
> 0).
Comment. If
, (which is equivalent N 1 N 2 > 0) then property P3 takes the form
(a
> 0, a
≠ 1, N
1
N
2
> 0).
P4. The logarithm of the power of a positive number is equal to the product of the exponent and the logarithm of this number:
log a N k = k log a N (a > 0, a ≠ 1, N > 0).
Comment. If k - even number (k = 2s), That
log a N 2s = 2s log a |N | (a > 0, a ≠ 1, N ≠ 0).
P5. Formula for moving to another base:
(a
> 0, a
≠ 1, b
> 0, b
≠ 1, N
> 0),
in particular if N = b, we get
(a > 0, a ≠ 1, b > 0, b ≠ 1). (2)Using properties P4 and P5, it is easy to obtain the following properties
(a
> 0, a
≠ 1, b
> 0, c
≠ 0), (3)
(a
> 0, a
≠ 1, b
> 0, c
≠ 0), (4)
(a
> 0, a
≠ 1, b
> 0, c
≠ 0), (5)
and, if in (5) c- even number ( c = 2n), occurs
(b
> 0, a
≠ 0, |a
| ≠ 1). (6)
Let us list the main properties of the logarithmic function f (x) = log a x :
1. The domain of definition of a logarithmic function is the set of positive numbers.
2. The range of values of the logarithmic function is the set of real numbers.
3. When a> 1 logarithmic function is strictly increasing (0< x 1 < x 2log a x 1 < loga x 2), and at 0< a < 1, - строго убывает (0 < x 1 < x 2log a x 1 > log a x 2).
4.log a 1 = 0 and log a a = 1 (a > 0, a ≠ 1).
5. If a> 1, then the logarithmic function is negative when x(0;1) and positive at x(1;+∞), and if 0< a < 1, то логарифмическая функция положительна при x (0;1) and negative at x (1;+∞).
6. If a> 1, then the logarithmic function is convex upward, and if a(0;1) - convex downwards.
The following statements (see, for example,) are used when solving logarithmic equations.
An inequality is called logarithmic if it contains a logarithmic function.
Solution methods logarithmic inequalities no different from , except for two things.
Firstly, when moving from the logarithmic inequality to the inequality of sublogarithmic functions, one should follow the sign of the resulting inequality. It obeys the following rule.
If the base of the logarithmic function is greater than $1$, then when moving from the logarithmic inequality to the inequality of sublogarithmic functions, the sign of the inequality is preserved, but if it is less than $1$, then it changes to the opposite.
Secondly, the solution to any inequality is an interval, and, therefore, at the end of solving the inequality of sublogarithmic functions it is necessary to create a system of two inequalities: the first inequality of this system will be the inequality of sublogarithmic functions, and the second will be the interval of the domain of definition of the logarithmic functions included in the logarithmic inequality.
Practice.
Let's solve the inequalities:
1. $\log_(2)((x+3)) \geq 3.$
$D(y): \x+3>0.$
$x \in (-3;+\infty)$
The base of the logarithm is $2>1$, so the sign does not change. Using the definition of logarithm, we get:
$x+3 \geq 2^(3),$
$x \in )