Graph of the natural logarithm function. Properties of natural logarithms: graph, base, functions, limit, formulas and domain of definition

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 (\displaystyle e)- an irrational constant equal to approximately 2.72. It is denoted as ln ⁡ x (\displaystyle \ln x), log e ⁡ x (\displaystyle \log _(e)x) or sometimes just log ⁡ x (\displaystyle \log x), if the base e (\displaystyle e) implied . In other words, the natural logarithm of a number x- this is an exponent to which a number must be raised e, To obtain x. This definition can be extended to complex numbers.

ln ⁡ e = 1 (\displaystyle \ln e=1), because e 1 = e (\displaystyle e^(1)=e); ln ⁡ 1 = 0 (\displaystyle \ln 1=0), because e 0 = 1 (\displaystyle e^(0)=1).

The natural logarithm can also be defined geometrically for any positive real number a as the area under the curve y = 1 x (\displaystyle y=(\frac (1)(x))) in between [ 1 ; a ] (\displaystyle ). The simplicity of this definition, which is consistent with many other formulas that use this logarithm, explains the origin of the name "natural".

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

e ln ⁡ a = a (a > 0) ; (\displaystyle e^(\ln a)=a\quad (a>0);) ln ⁡ e a = a (a > 0) . (\displaystyle \ln e^(a)=a\quad (a>0).)

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

ln ⁡ x y = ln ⁡ x + ln ⁡ y . (\displaystyle \ln xy=\ln x+\ln y.)

This could be, for example, a calculator from the basic set of operating room programs Windows systems. The link to launch it is hidden quite in the main menu of the OS - open it by clicking on the “Start” button, then open its “Programs” section, go to the “Standard” subsection, and then to the “Utilities” section and, finally, click on the “Calculator” item " Instead of using the mouse and navigating through menus, you can use the keyboard and the program launch dialog - press the WIN + R key combination, type calc (this is the name of the calculator executable file) and press Enter.

Switch the calculator interface to advanced mode, which allows you to do... By default it opens in “normal” view, but you need “engineering” or “ ” (depending on the version of the OS you are using). Expand the “View” section in the menu and select the appropriate line.

Enter the argument whose natural value you want to evaluate. This can be done either from the keyboard or by clicking the corresponding buttons in the calculator interface on the screen.

Click the button labeled ln - the program will calculate the logarithm to base e and show the result.

Use one of the -calculators as an alternative to calculating the value of the natural logarithm. For example, the one located at http://calc.org.ua. Its interface is extremely simple - there is a single input field where you need to type the value of the number, the logarithm of which you need to calculate. Among the buttons, find and click the one that says ln. The script of this calculator does not require sending data to the server and a response, so you will receive the calculation result almost instantly. The only feature that should be taken into account is the separator between the fractional and whole part The entered number must have a dot here, not a .

The term " logarithm"descended from two Greek words, one of which stands for "number" and the other for "ratio". It denotes the mathematical operation of calculating a variable quantity (exponent) to which a constant value (base) must be raised to obtain the number indicated under the sign logarithm A. If the base is equal to a mathematical constant called the number "e", then logarithm called "natural".

You will need

• Internet access, Microsoft Office Excel or calculator.

Instructions

Use the many calculators available on the Internet - this is perhaps an easy way to calculate natural a. You don’t have to search for the appropriate service, since many search engines and themselves have built-in calculators, quite suitable for working with logarithm ami. For example, go to the main page of the largest online search engine - Google. No buttons are required here to enter values ​​or select functions; just enter the desired mathematical action in the query input field. Let's say, to calculate logarithm and the number 457 in base “e”, enter ln 457 - this will be enough for Google to display with an accuracy of eight decimal places (6.12468339) even without pressing the button to send a request to the server.

Use the appropriate built-in function if you need to calculate the value of a natural logarithm and occurs when working with data in a popular spreadsheet Microsoft editor Office Excel. This function is called here using the common notation logarithm and in upper case - LN. Select the cell in which the calculation result should be displayed and enter an equal sign - this is how in this spreadsheet editor records should begin in the cells containing in the “Standard” subsection of the “All Programs” section of the main menu. Switch the calculator to a more functional mode by pressing Alt + 2. Then enter the value, natural logarithm which you want to calculate, and click in the program interface the button indicated by the symbols ln. The application will perform the calculation and display the result.

Video on the topic

The basic properties of the 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 are given.

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 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:

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

Logarithm 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 by 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 a decimal, enclosed between 0 and 1. The integer part 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 linear parts we find column 4. At the intersection of line 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.

Tables and Properties 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; the 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 the natural logarithms of numbers equal to the products of the number 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 cologarithms 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 four modern rules operations on exponents or four rules for operations on logarithms: the sum in the top line corresponds to the product in 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 of 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.

The first logarithms, for 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 bolder 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 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-log paper (where the y-axis has a logarithmic scale and the x-axis has a uniform scale) is useful when you need to identify exponential functions. Equations of the form y = kb rx occur whenever some quantity, such as population size, amount of radioactive material, or bank balance, decreases or increases at a rate proportional to the available this moment number of inhabitants, 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, mollusk shells are twisted Nautilus, horns mountain sheep and parrot beaks. All these natural forms 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 appears in such distant and completely various areas, like the contour of an eccentric cam and the trajectory of some insects flying towards the light.

Lesson and presentation on the topics: "Natural logarithms. The base of the natural logarithm. The logarithm of a natural number"

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What is natural logarithm

Guys, in the last lesson we learned a new, special number - e. Today we will continue to work with this number.
We have studied logarithms and we know that the base of a logarithm can be many numbers that are greater than 0. Today we will also look at a logarithm whose base is the number e. Such a logarithm is usually called the natural logarithm. It has its own notation: $\ln(n)$ is the natural logarithm. This entry is equivalent to the entry: $\log_e(n)=\ln(n)$.
Illustrative and logarithmic functions are inverse, then the natural logarithm is the inverse of the function: $y=e^x$.
Inverse functions are symmetric with respect to the straight line $y=x$.
Let's plot the natural logarithm by plotting the exponential function with respect to the straight line $y=x$.

It is worth noting that the angle of inclination of the tangent to the graph of the function $y=e^x$ at point (0;1) is 45°. Then the angle of inclination of the tangent to the graph of the natural logarithm at point (1;0) will also be equal to 45°. Both of these tangents will be parallel to the line $y=x$. Let's diagram the tangents:

Properties of the function $y=\ln(x)$

1. $D(f)=(0;+∞)$.
2. Is neither even nor odd.
3. Increases throughout the entire domain of definition.
4. Not limited from above, not limited from below.
5. Greatest value No, lowest value No.
6. Continuous.
7. $E(f)=(-∞; +∞)$.
8. Convex upward.
9. Differentiable everywhere.

I know higher mathematics it has been proven that the derivative of an inverse function is the inverse of the derivative of a given function.
There is not much point in going into the proof, let's just write the formula: $y"=(\ln(x))"=\frac(1)(x)$.

Example.
Calculate the value of the derivative of the function: $y=\ln(2x-7)$ at the point $x=4$.
Solution.
IN general view our function is represented by the function $y=f(kx+m)$, we can calculate the derivatives of such functions.
$y"=(\ln((2x-7)))"=\frac(2)((2x-7))$.
Let's calculate the value of the derivative at the required point: $y"(4)=\frac(2)((2*4-7))=2$.
Answer: 2.

Example.
Draw a tangent to the graph of the function $y=ln(x)$ at the point $х=е$.
Solution.
We remember well the equation of the tangent to the graph of a function at the point $x=a$.
$y=f(a)+f"(a)(x-a)$.
We sequentially calculate the required values.
$a=e$.
$f(a)=f(e)=\ln(e)=1$.
$f"(a)=\frac(1)(a)=\frac(1)(e)$.
$y=1+\frac(1)(e)(x-e)=1+\frac(x)(e)-\frac(e)(e)=\frac(x)(e)$.
The tangent equation at the point $x=e$ is the function $y=\frac(x)(e)$.
Let's plot the natural logarithm and the tangent line.

Example.
Examine the function for monotonicity and extrema: $y=x^6-6*ln(x)$.
Solution.
The domain of definition of the function $D(y)=(0;+∞)$.
Let's find the derivative of the given function:
$y"=6*x^5-\frac(6)(x)$.
The derivative exists for all x from the domain of definition, then there are no critical points. Let's find stationary points:
$6*x^5-\frac(6)(x)=0$.
$\frac(6*x^6-6)(x)=0$.
$6*x^6-6=0$.
$x^6-1=0$.
$x^6=1$.
$x=±1$.
The point $х=-1$ does not belong to the domain of definition. Then we have one stationary point $x=1$. Let's find the intervals of increasing and decreasing:

Point $x=1$ is the minimum point, then $y_min=1-6*\ln(1)=1$.
Answer: The function decreases on the segment (0;1], the function increases on the ray $)