What is the base of the natural logarithm. Understanding the natural logarithm

The main properties are given natural logarithm, graph, domain of definition, set of values, basic formulas, derivative, integral, power series expansion and representation of the function ln x using complex numbers.

Definition

Natural logarithm is the function y = ln x, the inverse of the exponential, x = e y, and is the logarithm to the base of the number e: ln x = log e x.

The natural logarithm is widely used in mathematics because its derivative has the simplest form: (ln x)′ = 1/ x.

Based on definitions, the base of the natural logarithm is the number e:
e ≅ 2.718281828459045...;
.

Graph of the function y = ln x.

Graph of natural logarithm (functions y = ln x) is obtained from the exponential graph by mirror reflection relative to the straight line y = x.

The natural logarithm is defined at positive values variable x. It increases monotonically in its domain of definition.

At x → 0 the limit of the natural logarithm is minus infinity (-∞).

As x → + ∞, the limit of the natural logarithm is plus infinity (+ ∞). For large x, the logarithm increases quite slowly. Any power function x a with a positive exponent a grows faster than the logarithm.

Properties of the natural logarithm

Domain of definition, set of values, extrema, increase, decrease

The natural logarithm is a monotonically increasing function, so it has no extrema. The main properties of the natural logarithm are presented in the table.

ln x values

ln 1 = 0

Basic formulas for natural logarithms

Formulas following from the definition of the inverse function:

The main property of logarithms and its consequences

Base replacement formula

Any logarithm can be expressed in terms of natural logarithms using the base substitution formula:

Proofs of these formulas are presented in the section "Logarithm".

Inverse function

The inverse of the natural logarithm is the exponent.

If , then

If, then.

Derivative ln x

Derivative of the natural logarithm:
.
Derivative of the natural logarithm of modulus x:
.
Derivative of nth order:
.
Deriving formulas > > >

Integral

The integral is calculated by integration by parts:
.
So,

Expressions using complex numbers

Consider the function of the complex variable z:
.
Let's express the complex variable z via module r and argument φ :
.
Using the properties of the logarithm, we have:
.
Or
.
The argument φ is not uniquely defined. If you put
, where n is an integer,
it will be the same number for different n.

Therefore, the natural logarithm, as a function of a complex variable, is not a single-valued function.

Power series expansion

When the expansion takes place:

Used literature:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.

Natural logarithm

Graph of the natural logarithm function. The function slowly approaches positive infinity as it increases x and quickly approaches negative infinity when x tends to 0 (“slow” and “fast” compared to any power function of x).

Natural logarithm is the logarithm to the base , Where e- an irrational constant equal to approximately 2.718281 828. The natural logarithm is usually written as ln( x), log e (x) or sometimes just log( x), if the base e implied.

Natural logarithm of a number x(written as ln(x)) is the exponent to which the number must be raised e to get x. For example, ln(7,389...) equals 2 because e 2 =7,389... . Natural logarithm of the number itself e (ln(e)) is equal to 1 because e 1 = e, and the natural logarithm is 1 ( ln(1)) is equal to 0 because e 0 = 1.

The natural logarithm can be defined for any positive real number a as the area under the curve y = 1/x from 1 to a. The simplicity of this definition, which is consistent with many other formulas that use the natural logarithm, led to the name "natural". This definition can be extended to complex numbers, as discussed below.

If we consider the natural logarithm as a real function of a real variable, then it is the inverse function of the exponential function, which leads to the identities:

Like all logarithms, the natural logarithm maps multiplication to addition:

Thus, the logarithmic function is an isomorphism of the group of positive real numbers with respect to multiplication by the group of real numbers with respect to addition, which can be represented as a function:

The logarithm can be defined for any positive base other than 1, not just e, but logarithms for other bases differ from the natural logarithm only by a constant factor, and are usually defined in terms of the natural logarithm. Logarithms are useful for solving equations that involve unknowns as exponents. For example, logarithms are used to find the decay constant for a known half-life, or to find the decay time in solving radioactivity problems. They play an important role in many areas of mathematics and applied sciences, and are used in finance to solve many problems, including finding compound interest.

Story

The first mention of the natural logarithm was made by Nicholas Mercator in his work Logarithmotechnia, published in 1668, although mathematics teacher John Spidell compiled a table of natural logarithms back in 1619. It was previously called the hyperbolic logarithm because it corresponds to the area under the hyperbola. It is sometimes called the Napier logarithm, although the original meaning of this term was somewhat different.

Designation conventions

The natural logarithm is usually denoted by “ln( x)", logarithm to base 10 - via "lg( x)", and other reasons are usually indicated explicitly with the symbol "log".

In many works on discrete mathematics, cybernetics, and computer science, authors use the notation “log( x)" for logarithms to base 2, but this convention is not generally accepted and requires clarification either in the list of notations used or (in the absence of such a list) by a footnote or comment when first used.

Parentheses around the argument of logarithms (if this does not lead to an erroneous reading of the formula) are usually omitted, and when raising a logarithm to a power, the exponent is assigned directly to the sign of the logarithm: ln 2 ln 3 4 x 5 = [ ln ( 3 )] 2 .

Anglo-American system

Mathematicians, statisticians and some engineers usually use the term “natural logarithm” or “log( x)" or "ln( x)", and to denote the base 10 logarithm - "log 10 ( x)».

Some engineers, biologists and other specialists always write “ln( x)" (or occasionally "log e ( x)") when they mean the natural logarithm, and the notation "log( x)" they mean log 10 ( x).

log e is a "natural" logarithm because it occurs automatically and appears very often in mathematics. For example, consider the problem of the derivative of a logarithmic function:

If the base b equals e, then the derivative is simply 1/ x, and when x= 1 this derivative is equal to 1. Another reason why the base e The most natural thing about the logarithm is that it can be defined quite simply in terms of a simple integral or Taylor series, which cannot be said about other logarithms.

Further justifications for naturalness are not related to notation. For example, there are several simple series with natural logarithms. Pietro Mengoli and Nicholas Mercator called them logarithmus naturalis several decades until Newton and Leibniz developed differential and integral calculus.

Definition

Formally ln( a) can be defined as the area under the curve of the graph 1/ x from 1 to a, i.e. as an integral:

It is truly a logarithm because it satisfies the fundamental property of the logarithm:

This can be demonstrated by assuming as follows:

Numerical value

To calculate the numerical value of the natural logarithm of a number, you can use its Taylor series expansion in the form:

To get better speed convergence, we can use the following identity:

provided that y = (x−1)/(x+1) and x > 0.

For ln( x), Where x> 1, the closer the value x to 1, then faster speed convergence. The identities associated with the logarithm can be used to achieve the goal:

These methods were used even before the advent of calculators, for which numerical tables were used and manipulations similar to those described above were performed.

High accuracy

To calculate the natural logarithm with a large number accuracy numbers, the Taylor series is not efficient because its convergence is slow. An alternative is to use Newton's method to invert into an exponential function whose series converges more quickly.

An alternative for very high calculation accuracy is the formula:

Where M denotes the arithmetic-geometric average of 1 and 4/s, and

m chosen so that p marks of accuracy is achieved. (In most cases, a value of 8 for m is sufficient.) In fact, if this method is used, Newton's inverse of the natural logarithm can be applied to efficiently calculate the exponential function. (The constants ln 2 and pi can be pre-calculated to the desired accuracy using any of the known rapidly convergent series.)

Computational complexity

The computational complexity of natural logarithms (using the arithmetic-geometric mean) is O( M(n)ln n). Here n is the number of digits of precision for which the natural logarithm must be evaluated, and M(n) is the computational complexity of multiplying two n-digit numbers.

Continued fractions

Although there are no simple continued fractions to represent a logarithm, several generalized continued fractions can be used, including:

Complex logarithms

The exponential function can be extended to a function that gives a complex number of the form e x for any arbitrary complex number x, in this case an infinite series with complex x. This exponential function can be inverted to form a complex logarithm, which will have most of the properties of ordinary logarithms. There are, however, two difficulties: there is no x, for which e x= 0, and it turns out that e 2πi = 1 = e 0 . Since the multiplicativity property is valid for a complex exponential function, then e z = e z+2nπi for all complex z and whole n.

The logarithm cannot be defined over the entire complex plane, and even so it is multivalued - any complex logarithm can be replaced by an "equivalent" logarithm by adding any integer multiple of 2 πi. The complex logarithm can only be single-valued on a slice of the complex plane. For example, ln i = 1/2 πi or 5/2 πi or −3/2 πi, etc., and although i 4 = 1.4 log i can be defined as 2 πi, or 10 πi or −6 πi, and so on.

Notes

Mathematics for physical chemistry. - 3rd. - Academic Press, 2005. - P. 9. - ISBN 0-125-08347-5,Extract of page 9
J J O"Connor and E F Robertson The number e. The MacTutor History of Mathematics archive (September 2001). Archived
Cajori Florian A History of Mathematics, 5th ed. - AMS Bookstore, 1991. - P. 152. - ISBN 0821821024
Flashman, Martin Estimating Integrals using Polynomials. Archived from the original on February 12, 2012.

often take a number e = 2,718281828 . Logarithms based on this base are called natural. When performing calculations with natural logarithms, it is common to operate with the sign ln, not log; while the number 2,718281828 , defining the basis, are not indicated.

In other words, the formulation will look like: natural logarithm numbers X- this is an exponent to which a number must be raised e to get x.

So, ln(7,389...)= 2, since e 2 =7,389... . Natural logarithm of the number itself e= 1 because e 1 =e, and the natural logarithm of unity is zero, since e 0 = 1.

The number itself e defines the limit of a monotone bounded sequence

it is calculated that e = 2,7182818284... .

Quite often, in order to fix a number in memory, the digits of the required number are associated with some outstanding date. Speed of memorizing the first nine digits of a number e after the decimal point will increase if you notice that 1828 is the year of birth of Leo Tolstoy!

Today there are enough full tables natural logarithms.

Natural logarithm graph(functions y =ln x) is a consequence of the exponential graph as a mirror image of the straight line y = x and has the form:

The natural logarithm can be found for every positive real number a as the area under the curve y = 1/x from 1 to a.

The elementary nature of this formulation, which is consistent with many other formulas in which the natural logarithm is involved, was the reason for the formation of the name “natural”.

If you analyze natural logarithm, as a real function of a real variable, then it acts inverse function to an exponential function, which reduces to the identities:

e ln(a) =a (a>0)

ln(e a) =a

By analogy with all logarithms, the natural logarithm converts multiplication into addition, division into subtraction:

ln(xy) = ln(x) + ln(y)

ln(x/y)= lnx - lny

The logarithm can be found for every positive base that is not equal to one, not just for e, but logarithms for other bases differ from the natural logarithm only by a constant factor, and are usually defined in terms of the natural logarithm.

Having analyzed natural logarithm graph, we find that it exists for positive values of the variable x. It increases monotonically in its domain of definition.

At x → 0 the limit of the natural logarithm is minus infinity ( -∞ ).At x → +∞ the limit of the natural logarithm is plus infinity ( + ∞ ). At large x The logarithm increases quite slowly. Any power function xa with a positive exponent a increases faster than the logarithm. The natural logarithm is a monotonically increasing function, so it has no extrema.

Usage natural logarithms very rational when passing higher mathematics. Thus, using the logarithm is convenient for finding the answer to equations in which unknowns appear as exponents. The use of natural logarithms in calculations makes it possible to greatly simplify large number mathematical formulas. Logarithms to the base e are present in solving a significant number of physical problems and naturally are included in the mathematical description of individual chemical, biological and other processes. Thus, logarithms are used to calculate the decay constant for a known half-life, or to calculate the decay time in solving problems of radioactivity. They perform in leading role in many branches of mathematics and practical sciences, they are resorted to in the field of finance to solve large number tasks, including the calculation of compound interest.

Logarithm of a given number is called the exponent to which another number must be raised, called basis logarithm to get this number. For example, the base 10 logarithm of 100 is 2. In other words, 10 must be squared to get 100 (10 2 = 100). If n– a given number, b– base and l– logarithm, then b l = n. Number n also called base antilogarithm b numbers l. For example, the antilogarithm of 2 to base 10 is equal to 100. This can be written in the form of the relations log b n = l and antilog b l = n.

Basic properties of logarithms:

Any positive number other than one can serve as a base for logarithms, but unfortunately it turns out that if b And n are rational numbers, then in rare cases there is such a rational number l, What b l = n. However, it is possible to define an irrational number l, for example, such that 10 l= 2; this is an irrational number l can be approximated with any required accuracy rational numbers. It turns out that in the given example l is approximately equal to 0.3010, and this approximation of the base 10 logarithm of 2 can be found in four-digit tables of decimal logarithms. Base 10 logarithms (or base 10 logarithms) are so commonly used in calculations that they are called ordinary logarithms and written as log2 = 0.3010 or log2 = 0.3010, omitting the explicit indication of the base of the logarithm. Logarithms to the base e, a transcendental number approximately equal to 2.71828, are called natural logarithms. They are found mainly in works on mathematical analysis and its applications to various sciences. Natural logarithms are also written without explicitly indicating the base, but using the special notation ln: for example, ln2 = 0.6931, because e 0,6931 = 2.

Using tables of ordinary logarithms.

The regular logarithm of a number is an exponent to which 10 must be raised to obtain a given number. Since 10 0 = 1, 10 1 = 10 and 10 2 = 100, we immediately get that log1 = 0, log10 = 1, log100 = 2, etc. for increasing integer powers 10. Likewise, 10 –1 = 0.1, 10 –2 = 0.01 and therefore log0.1 = –1, log0.01 = –2, etc. for all integers negative powers 10. The usual logarithms of the remaining numbers are contained between the logarithms of the nearest integer powers of the number 10; log2 must be between 0 and 1, log20 must be between 1 and 2, and log0.2 must be between -1 and 0. Thus, the logarithm consists of two parts, an integer and decimal, enclosed between 0 and 1. The integer part is called characteristic logarithm and is determined by the number itself, the fractional part is called mantissa and can be found from tables. Also, log20 = log(2ґ10) = log2 + log10 = (log2) + 1. The logarithm of 2 is 0.3010, so log20 = 0.3010 + 1 = 1.3010. Similarly, log0.2 = log(2о10) = log2 – log10 = (log2) – 1 = 0.3010 – 1. After subtraction, we get log0.2 = – 0.6990. However, it is more convenient to represent log0.2 as 0.3010 – 1 or as 9.3010 – 10; can be formulated and general rule: all numbers obtained from a given number by multiplication by a power of 10 have the same mantissa, equal to the mantissa of the given number. Most tables show the mantissas of numbers in the range from 1 to 10, since the mantissas of all other numbers can be obtained from those given in the table.

Most tables give logarithms with four or five decimal places, although there are seven-digit tables and tables with even more decimal places. The easiest way to learn how to use such tables is with examples. To find log3.59, first of all, we note that the number 3.59 is between 10 0 and 10 1, so its characteristic is 0. We find the number 35 (on the left) in the table and move along the row to the column that has the number 9 at the top ; the intersection of this column and row 35 is 5551, so log3.59 = 0.5551. To find the mantissa of a number with four significant figures, it is necessary to resort to interpolation. In some tables, interpolation is facilitated by the proportions given in the last nine columns on the right side of each page of the tables. Let us now find log736.4; the number 736.4 lies between 10 2 and 10 3, therefore the characteristic of its logarithm is 2. In the table we find a row to the left of which there is 73 and column 6. At the intersection of this row and this column there is the number 8669. Among the linear parts we find column 4 . At the intersection of row 73 and column 4 there is the number 2. By adding 2 to 8669, we get the mantissa - it is equal to 8671. Thus, log736.4 = 2.8671.

Natural logarithms.

The tables and properties of natural logarithms are similar to the tables and properties of ordinary logarithms. The main difference between both is that the integer part of the natural logarithm is not significant in determining the position of the decimal point, and therefore the difference between the mantissa and the characteristic does not play a special role. Natural logarithms of numbers 5.432; 54.32 and 543.2 are equal to 1.6923, respectively; 3.9949 and 6.2975. The relationship between these logarithms will become obvious if we consider the differences between them: log543.2 – log54.32 = 6.2975 – 3.9949 = 2.3026; last number is nothing more than the natural logarithm of the number 10 (written like this: ln10); log543.2 – log5.432 = 4.6052; the last number is 2ln10. But 543.2 = 10ґ54.32 = 10 2ґ5.432. Thus, by the natural logarithm of a given number a you can find natural logarithms of numbers, equal to the products numbers a for any degree n numbers 10 if to ln a add ln10 multiplied by n, i.e. ln( aґ10n) = log a + n ln10 = ln a + 2,3026n. For example, ln0.005432 = ln(5.432ґ10 –3) = ln5.432 – 3ln10 = 1.6923 – (3ґ2.3026) = – 5.2155. Therefore, tables of natural logarithms, like tables of ordinary logarithms, usually contain only logarithms of numbers from 1 to 10. In the system of natural logarithms, one can talk about antilogarithms, but more often they talk about an exponential function or an exponent. If x= log y, That y = e x, And y called the exponent of x(for typographic convenience, they often write y= exp x). The exponent plays the role of the antilogarithm of the number x.

Using tables of decimal and natural logarithms, you can create tables of logarithms in any base other than 10 and e. If log b a = x, That b x = a, and therefore log c b x= log c a or x log c b= log c a, or x= log c a/log c b= log b a. Therefore, using this inversion formula from the base logarithm table c you can build tables of logarithms in any other base b. Multiplier 1/log c b called transition module from the base c to the base b. Nothing prevents, for example, using the inversion formula or transition from one system of logarithms to another, finding natural logarithms from the table of ordinary logarithms or making the reverse transition. For example, log105.432 = log e 5.432/log e 10 = 1.6923/2.3026 = 1.6923ґ0.4343 = 0.7350. The number 0.4343, by which the natural logarithm of a given number must be multiplied to obtain an ordinary logarithm, is the modulus of the transition to the system of ordinary logarithms.

Special tables.

Logarithms were originally invented so that, using their properties log ab= log a+ log b and log a/b= log a– log b, turn products into sums and quotients into differences. In other words, if log a and log b are known, then using addition and subtraction we can easily find the logarithm of the product and the quotient. In astronomy, however, often given values of log a and log b need to find log( a + b) or log( a – b). Of course, one could first find from tables of logarithms a And b, then perform the indicated addition or subtraction and, again referring to the tables, find the required logarithms, but such a procedure would require referring to the tables three times. Z. Leonelli in 1802 published tables of the so-called. Gaussian logarithms– logarithms for adding sums and differences – which made it possible to limit oneself to one access to tables.

In 1624, I. Kepler proposed tables of proportional logarithms, i.e. logarithms of numbers a/x, Where a– some positive constant value. These tables are used primarily by astronomers and navigators.

Proportional logarithms at a= 1 are called by logarithms and are used in calculations when one has to deal with products and quotients. Cologarithm of a number n equal to the logarithm reciprocal number; those. colog n= log1/ n= – log n. If log2 = 0.3010, then colog2 = – 0.3010 = 0.6990 – 1. The advantage of using cologarithms is that when calculating the value of the logarithm of expressions like pq/r triple sum of positive decimals log p+ log q+colog r is easier to find than the mixed sum and difference log p+ log q– log r.

Story.

The principle underlying any system of logarithms has been known for a very long time and can be traced back to ancient Babylonian mathematics (circa 2000 BC). In those days, interpolation between table values of positive integer powers of integers was used to calculate compound interest. Much later, Archimedes (287–212 BC) used powers of 108 to find an upper limit on the number of grains of sand required to completely fill the then known Universe. Archimedes drew attention to the property of exponents that underlies the effectiveness of logarithms: the product of powers corresponds to the sum of the exponents. At the end of the Middle Ages and the beginning of the modern era, mathematicians increasingly began to turn to the relationship between geometric and arithmetic progressions. M. Stiefel in his essay Integer Arithmetic(1544) gave a table of positive and negative powers of the number 2:

Stiefel noticed that the sum of the two numbers in the first row (the exponent row) is equal to the exponent of two corresponding to the product of the two corresponding numbers in the bottom row (the exponent row). In connection with this table, Stiefel formulated four rules equivalent to the four modern rules for operations on exponents or the four rules for operations on logarithms: the sum on the top line corresponds to the product on the bottom line; subtraction on the top line corresponds to division on the bottom line; multiplication on the top line corresponds to exponentiation on the bottom line; division on the top line corresponds to rooting on the bottom line.

Apparently, rules similar to Stiefel’s rules led J. Naper to formally introduce the first system of logarithms in his work Description of the amazing table of logarithms, published in 1614. But Napier’s thoughts were occupied with the problem of converting products into sums ever since, more than ten years before the publication of his work, Napier received news from Denmark that at the Tycho Brahe Observatory his assistants had a method that made it possible to convert products into sums. The method mentioned in the message Napier received was based on the use trigonometric formulas type

therefore Naper's tables consisted mainly of logarithms trigonometric functions. Although the concept of base was not explicitly included in the definition proposed by Napier, the role equivalent to the base of the system of logarithms in his system was played by the number (1 – 10 –7)ґ10 7, approximately equal to 1/ e.

Independently of Naper and almost simultaneously with him, a system of logarithms, quite similar in type, was invented and published by J. Bürgi in Prague, published in 1620 Arithmetic and geometric progression tables. These were tables of antilogarithms to the base (1 + 10 –4) ґ10 4, a fairly good approximation of the number e.

In the Naper system, the logarithm of the number 10 7 was taken to be zero, and as the numbers decreased, the logarithms increased. When G. Briggs (1561–1631) visited Napier, both agreed that it would be more convenient to use the number 10 as the base and consider the logarithm of one to be zero. Then, as the numbers increased, their logarithms would increase. So we got modern system decimal logarithms, a table of which Briggs published in his work Logarithmic arithmetic(1620). Logarithms to the base e, although not exactly those introduced by Naper, are often called Naper's. The terms "characteristic" and "mantissa" were proposed by Briggs.

First logarithms in force historical reasons used approximations to the numbers 1/ e And e. Somewhat later, the idea of natural logarithms began to be associated with the study of areas under a hyperbola xy= 1 (Fig. 1). In the 17th century it was shown that the area bounded by this curve, the axis x and ordinates x= 1 and x = a(in Fig. 1 this area is covered with thicker and sparse dots) increases in arithmetic progression, When a increases exponentially. It is precisely this dependence that arises in the rules for operations with exponents and logarithms. This gave rise to calling Naperian logarithms “hyperbolic logarithms.”

Logarithmic function.

There was a time when logarithms were considered solely as a means of calculation, but in the 18th century, mainly thanks to the work of Euler, the concept of a logarithmic function was formed. Graph of such a function y= log x, whose ordinates increase in an arithmetic progression, while the abscissas increase in a geometric progression, is presented in Fig. 2, A. Graph of an inverse or exponential function y = e x, whose ordinates increase in geometric progression, and whose abscissas increase in arithmetic progression, is presented, respectively, in Fig. 2, b. (Curves y= log x And y = 10x similar in shape to curves y= log x And y = e x.) Alternative definitions of the logarithmic function have also been proposed, e.g.

kpi ; and, similarly, the natural logarithms of the number -1 are complex numbers types (2 k + 1)pi, Where k– an integer. Similar statements are true for general logarithms or other systems of logarithms. Additionally, the definition of logarithms can be generalized using Euler's identities to include complex logarithms of complex numbers.

An alternative definition of a logarithmic function is provided by functional analysis. If f(x) – continuous function of a real number x, having the following three properties: f (1) = 0, f (b) = 1, f (uv) = f (u) + f (v), That f(x) is defined as the logarithm of the number x based on b. This definition has a number of advantages over the definition given at the beginning of this article.

Applications.

Logarithms were originally used solely to simplify calculations, and this application is still one of their most important. The calculation of products, quotients, powers and roots is facilitated not only by the wide availability of published tables of logarithms, but also by the use of so-called. slide rule - a computational tool whose operating principle is based on the properties of logarithms. The ruler is equipped with logarithmic scales, i.e. distance from number 1 to any number x chosen to be equal to log x; By shifting one scale relative to another, it is possible to plot the sums or differences of logarithms, which makes it possible to read directly from the scale the products or quotients of the corresponding numbers. You can also take advantage of the advantages of representing numbers in logarithmic form. logarithmic paper for plotting graphs (paper with logarithmic scales printed on it on both coordinate axes). If a function satisfies a power law of the form y = kxn, then its logarithmic graph looks like a straight line, because log y=log k + n log x– equation linear with respect to log y and log x. On the contrary, if the logarithmic graph of some functional dependence looks like a straight line, then this dependence is a power one. Semi-logarithmic paper (in which the y-axis has a logarithmic scale and the abscissa axis has a uniform scale) is convenient in cases where you need to identify exponential functions. Equations of the form y = kb rx occur whenever a quantity, such as a population, a quantity of radioactive material, or a bank balance, decreases or increases at a rate proportional to the available at the moment number of residents, radioactive substance or money. If such a dependence is plotted on semi-logarithmic paper, the graph will look like a straight line.

The logarithmic function arises in connection with a wide variety of natural forms. Flowers in sunflower inflorescences are arranged in logarithmic spirals, and mollusk shells are twisted. Nautilus, horns mountain sheep and parrot beaks. All of these natural shapes can serve as examples of a curve known as a logarithmic spiral because, in a polar coordinate system, its equation is r = ae bq, or ln r= log a + bq. Such a curve is described by a moving point, the distance from the pole of which increases in geometric progression, and the angle described by its radius vector increases in arithmetic progression. The ubiquity of such a curve, and therefore of the logarithmic function, is well illustrated by the fact that it occurs in such distant and completely different areas as the contour of an eccentric cam and the trajectory of some insects flying towards the light.

The logarithm of a number b to base a is the exponent to which the number a must be raised to obtain the number b.

If, then.

Logarithm - extreme important mathematical quantity, since logarithmic calculus allows not only to solve exponential equations, but also operate with indicators, differentiate exponential and logarithmic functions, integrate them and lead to a more acceptable form to be calculated.

All properties of logarithms are directly related to the properties exponential functions. For example, the fact that means that:

It should be noted that when solving specific problems, the properties of logarithms may turn out to be more important and useful than the rules for working with powers.

Let us present some identities:

Here are the basic algebraic expressions:

;

Attention! can exist only for x>0, x≠1, y>0.

Let's try to understand the question of what natural logarithms are. Special interest in mathematics represent two types- the first one has the number “10” at the base, and is called “ decimal logarithm" The second one is called natural. The base of the natural logarithm is the number “e”. This is what we will talk about in detail in this article.

Designations:

lg x - decimal;
ln x - natural.

Using the identity, we can see that ln e = 1, as well as the fact that lg 10=1.

Natural logarithm graph

Let's construct a graph of the natural logarithm using the standard classical method point by point. If you wish, you can check whether we are constructing the function correctly by examining the function. However, it makes sense to learn how to build it “manually” in order to know how to correctly calculate the logarithm.

Function: y = ln x. Let's write down a table of points through which the graph will pass:

Let us explain why we chose these particular values of the argument x. It's all about identity: . For the natural logarithm this identity will look like this:

For convenience, we can take five reference points:

;

Thus, calculating natural logarithms is a fairly simple task; moreover, it simplifies calculations of operations with powers, turning them into ordinary multiplication.

By plotting a graph point by point, we get an approximate graph:

The domain of definition of the natural logarithm (i.e. all valid values argument X) - all numbers are greater than zero.

Attention! The domain of definition of the natural logarithm includes only positive numbers! The scope of definition does not include x=0. This is impossible based on the conditions for the existence of the logarithm.

The range of values (i.e. all valid values of the function y = ln x) is all numbers in the interval.

Natural log limit

Studying the graph, the question arises - how does the function behave at y<0.

Obviously, the graph of the function tends to cross the y-axis, but will not be able to do this, since the natural logarithm at x<0 не существует.

Limit of natural log can be written this way:

Formula for replacing the base of a logarithm

Dealing with a natural logarithm is much easier than dealing with a logarithm that has an arbitrary base. That is why we will try to learn how to reduce any logarithm to a natural one, or express it to an arbitrary base through natural logarithms.

Let's start with the logarithmic identity:

Then any number or variable y can be represented as:

where x is any number (positive according to the properties of the logarithm).

This expression can be taken logarithmically on both sides. Let's do this using an arbitrary base z:

Let’s use the property (only instead of “c” we have the expression):

From here we get the universal formula:

In particular, if z=e, then:

We were able to represent a logarithm to an arbitrary base through the ratio of two natural logarithms.

We solve problems

In order to better understand natural logarithms, let's look at examples of several problems.

Problem 1. It is necessary to solve the equation ln x = 3.

Solution: Using the definition of the logarithm: if , then , we get:

Problem 2. Solve the equation (5 + 3 * ln (x - 3)) = 3.

Solution: Using the definition of the logarithm: if , then , we get:

Let's use the definition of a logarithm again:

Thus:

You can approximately calculate the answer, or you can leave it in this form.

Task 3. Solve the equation.

Solution: Let's make a substitution: t = ln x. Then the equation will take the following form:

We have a quadratic equation. Let's find its discriminant:

First root of the equation:

Second root of the equation:

Remembering that we made the substitution t = ln x, we get:

In statistics and probability theory, logarithmic quantities are found very often. This is not surprising, because the number e often reflects the growth rate of exponential quantities.

In computer science, programming and computer theory, logarithms are found quite often, for example, in order to store N bits in memory.

In the theories of fractals and dimensions, logarithms are constantly used, since the dimensions of fractals are determined only with their help.

In mechanics and physics There is no section where logarithms were not used. Barometric distribution, all the principles of statistical thermodynamics, the Tsiolkovsky equation, etc. are processes that can be mathematically described only using logarithms.

In chemistry, logarithms are used in Nernst equations and descriptions of redox processes.

Amazingly, even in music, in order to find out the number of parts of an octave, logarithms are used.

Natural logarithm Function y=ln x its properties

Proof of the main property of the natural logarithm