Natural logarithm, function ln x. Logarithm

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 mirror image relative to the straight line y = x.

The natural logarithm is defined for positive values ​​of the variable x.

It increases monotonically in its domain of definition. 0 At x →

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 φ :
.
and argument
.
Using the properties of the logarithm, we have:
.
Or
The argument φ is not uniquely defined. If you put
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.

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 obtain x. For example, ln(7,389...) is equal to 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 the 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 writing "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 obtain better speed convergence, we can use the following identity:

For ln( x), Where x> 1 than closer value x to 1, the faster the convergence rate. 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 big amount 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 Let's express the complex variable = e Let's express the complex variable+2nπi for all complex Let's express the complex variable 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.

see also

• John Napier - inventor of logarithms

Notes

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

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 number 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 that the separator between the fractional and integer parts of the entered number must be a dot, and not .

The term " logarithm" comes from two Greek words, one meaning "number" and the other meaning "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 home page of the largest online search engine - Google. No buttons for entering values ​​or selecting functions are required here, just type the desired one in the request input field mathematical operation. 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 the popular spreadsheet editor Microsoft Office Excel. This function is called here using the common notation such 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 logarithm of a positive number b to base a (a>0, a is not equal to 1) is a number c such that a c = b: log a b = c ⇔ a c = b (a > 0, a ≠ 1, b > 0)       

Note that the logarithm of a non-positive number is undefined. In addition, the base of the logarithm must be a positive number that is not equal to 1. For example, if we square -2, we get the number 4, but this does not mean that the base -2 logarithm of 4 is equal to 2.

Basic logarithmic identity

It is important that the scope of definition of the right and left sides of this formula is different. Left side is defined only for b>0, a>0 and a ≠ 1. The right-hand side is defined for any b, and does not depend on a at all. Thus, the application of the basic logarithmic “identity” when solving equations and inequalities can lead to a change in the OD.

Two obvious consequences of the definition of logarithm

Indeed, when raising the number a to the first power, we get the same number, and when raising it to the zero power, we get one.

Logarithm of the product and logarithm of the quotient

Log a b c = log a b − log a c (a > 0, a ≠ 1, b > 0, c > 0) (6)

I would like to warn schoolchildren against thoughtlessly using these formulas when solving logarithmic equations and inequalities. When using them “from left to right,” the ODZ narrows, and when moving from the sum or difference of logarithms to the logarithm of the product or quotient, the ODZ expands.

Indeed, the expression log a (f (x) g (x)) is defined in two cases: when both functions are strictly positive or when f (x) and g (x) are both less than zero.

Transforming this expression into the sum log a f (x) + log a g (x), we are forced to limit ourselves only to the case when f(x)>0 and g(x)>0. There is a narrowing of the range of acceptable values, and this is categorically unacceptable, since it can lead to a loss of solutions. A similar problem exists for formula (6).

The degree can be taken out of the sign of the logarithm

And again I would like to call for accuracy. Consider the following example:

Log a (f (x) 2 = 2 log a f (x)

The left side of the equality is obviously defined for all values ​​of f(x) except zero. The right side is only for f(x)>0! By taking the degree out of the logarithm, we again narrow the ODZ. The reverse procedure leads to an expansion of the range of acceptable values. All these remarks apply not only to power 2, but also to any even power.

Formula for moving to a new foundation

That rare case when the ODZ does not change during transformation. If you have chosen base c wisely (positive and not equal to 1), the formula for moving to a new base is completely safe.

If we choose the number b as the new base c, we get an important special case formulas (8):

Log a b = 1 log b a (a > 0, a ≠ 1, b > 0, b ≠ 1) (9)

Some simple examples with logarithms

Example 1. Calculate: log2 + log50.
Solution. log2 + log50 = log100 = 2. We used the sum of logarithms formula (5) and the definition of the decimal logarithm.

Example 2. Calculate: lg125/lg5.
Solution. log125/log5 = log 5 125 = 3. We used the formula for moving to a new base (8).

Table of formulas related to logarithms

As you know, when multiplying expressions with powers, their exponents always add up (a b *a c = a b+c). This mathematical law was derived by Archimedes, and later, in the 8th century, the mathematician Virasen created a table of integer exponents. It was they who served for the further discovery of logarithms. Examples of using this function can be found almost everywhere where you need to simplify cumbersome multiplication by simple addition. If you spend 10 minutes reading this article, we will explain to you what logarithms are and how to work with them. In simple and accessible language.

Definition in mathematics

A logarithm is an expression of the following form: log a b=c, that is, the logarithm of any non-negative number (that is, any positive) “b” to its base “a” is considered to be the power “c” to which the base “a” must be raised in order to ultimately get the value "b". Let's analyze the logarithm using examples, let's say there is an expression log 2 8. How to find the answer? It’s very simple, you need to find a power such that from 2 to the required power you get 8. After doing some calculations in your head, we get the number 3! And that’s true, because 2 to the power of 3 gives the answer as 8.

Types of logarithms

For many pupils and students, this topic seems complicated and incomprehensible, but in fact logarithms are not so scary, the main thing is to understand their general meaning and remember their properties and some rules. There are three separate types of logarithmic expressions:

1. Natural logarithm ln a, where the base is the Euler number (e = 2.7).
2. Decimal a, where the base is 10.
3. Logarithm of any number b to base a>1.

Each of them is solved in a standard way, including simplification, reduction and subsequent reduction to a single logarithm using logarithmic theorems. To obtain the correct values ​​of logarithms, you should remember their properties and the sequence of actions when solving them.

Rules and some restrictions

In mathematics, there are several rules-constraints that are accepted as an axiom, that is, they are not subject to discussion and are the truth. For example, it is impossible to divide numbers by zero, and it is also impossible to extract the even root of negative numbers. Logarithms also have their own rules, following which you can easily learn to work even with long and capacious logarithmic expressions:

• The base “a” must always be greater than zero, and not equal to 1, otherwise the expression will lose its meaning, because “1” and “0” to any degree are always equal to their values;
• if a > 0, then a b >0, it turns out that “c” must also be greater than zero.

How to solve logarithms?

For example, the task is given to find the answer to the equation 10 x = 100. This is very easy, you need to choose a power by raising the number ten to which we get 100. This, of course, is 10 2 = 100.

Now let's represent this expression in logarithmic form. We get log 10 100 = 2. When solving logarithms, all actions practically converge to find the power to which it is necessary to enter the base of the logarithm in order to obtain a given number.

To accurately determine the value unknown degree you need to learn how to work with the table of degrees. It looks like this:

As you can see, some exponents can be guessed intuitively if you have a technical mind and knowledge of the multiplication table. However, for larger values ​​you will need a power table. It can be used even by those who know nothing at all about complex mathematical topics. The left column contains numbers (base a), the top row of numbers is the value of the power c to which the number a is raised. At the intersection, the cells contain the number values ​​that are the answer (a c =b). Let's take, for example, the very first cell with the number 10 and square it, we get the value 100, which is indicated at the intersection of our two cells. Everything is so simple and easy that even the most true humanist will understand!

Equations and inequalities

It turns out that under certain conditions the exponent is the logarithm. Therefore, any mathematical numerical expressions can be written as a logarithmic equality. For example, 3 4 =81 can be written as the base 3 logarithm of 81 equal to four (log 3 81 = 4). For negative powers the rules are the same: 2 -5 = 1/32 we write it as a logarithm, we get log 2 (1/32) = -5. One of the most fascinating sections of mathematics is the topic of “logarithms”. We will look at examples and solutions of equations below, immediately after studying their properties. Now let's look at what inequalities look like and how to distinguish them from equations.

The following expression is given: log 2 (x-1) > 3 - it is a logarithmic inequality, since the unknown value “x” is under the logarithmic sign. And also in the expression two quantities are compared: the logarithm of the desired number to base two is greater than the number three.

The most important difference between logarithmic equations and inequalities is that equations with logarithms (for example, the logarithm 2 x = √9) imply one or more specific answers. numerical values, while when solving the inequality, both the range of permissible values ​​and the breakpoints of this function are determined. As a consequence, the answer is not a simple set of individual numbers, as in the answer to an equation, but a continuous series or set of numbers.

Basic theorems about logarithms

When solving primitive tasks of finding the values ​​of the logarithm, its properties may not be known. However, when it comes to logarithmic equations or inequalities, first of all, it is necessary to clearly understand and apply in practice all the basic properties of logarithms. We will look at examples of equations later; let's first look at each property in more detail.

1. The main identity looks like this: a logaB =B. It applies only when a is greater than 0, not equal to one, and B is greater than zero.
2. The logarithm of the product can be represented in the following formula: log d (s 1 * s 2) = log d s 1 + log d s 2. In this case, the mandatory condition is: d, s 1 and s 2 > 0; a≠1. You can give a proof for this logarithmic formula, with examples and solution. Let log a s 1 = f 1 and log a s 2 = f 2, then a f1 = s 1, a f2 = s 2. We obtain that s 1 * s 2 = a f1 *a f2 = a f1+f2 (properties of degrees ), and then by definition: log a (s 1 * s 2) = f 1 + f 2 = log a s1 + log a s 2, which is what needed to be proven.
3. The logarithm of the quotient looks like this: log a (s 1/ s 2) = log a s 1 - log a s 2.
4. The theorem in the form of a formula takes the following form: log a q b n = n/q log a b.

This formula is called the “property of the degree of logarithm.” It resembles the properties of ordinary degrees, and it is not surprising, because all mathematics is based on natural postulates. Let's look at the proof.

Let log a b = t, it turns out a t =b. If we raise both parts to the power m: a tn = b n ;

but since a tn = (a q) nt/q = b n, therefore log a q b n = (n*t)/t, then log a q b n = n/q log a b. The theorem has been proven.

Examples of problems and inequalities

The most common types of problems on logarithms are examples of equations and inequalities. They are found in almost all problem books, and are also a required part of mathematics exams. To enter a university or pass entrance examinations in mathematics, you need to know how to correctly solve such tasks.

Unfortunately, there is no single plan or scheme for solving and determining the unknown value of the logarithm, however, it can be applied to every mathematical inequality or logarithmic equation certain rules. First of all, you should find out whether the expression can be simplified or lead to general appearance. You can simplify long logarithmic expressions if you use their properties correctly. Let's get to know them quickly.

When solving logarithmic equations, we must determine what type of logarithm we have: an example expression may contain a natural logarithm or a decimal one.

Here are examples ln100, ln1026. Their solution boils down to the fact that they need to determine the power to which the base 10 will be equal to 100 and 1026, respectively. To solve natural logarithms, you need to apply logarithmic identities or their properties. Let's look at examples of solving logarithmic problems of various types.

How to Use Logarithm Formulas: With Examples and Solutions

So, let's look at examples of using the basic theorems about logarithms.

1. The property of the logarithm of a product can be used in tasks where it is necessary to expand great importance numbers b into simpler factors. For example, log 2 4 + log 2 128 = log 2 (4*128) = log 2 512. The answer is 9.
2. log 4 8 = log 2 2 2 3 = 3/2 log 2 2 = 1.5 - as you can see, using the fourth property of the logarithm power, we managed to solve a seemingly complex and unsolvable expression. You just need to factor the base and then take the exponent values ​​out of the sign of the logarithm.

Assignments from the Unified State Exam

Logarithms are often found in entrance exams, especially a lot of logarithmic problems in the Unified State Exam (state exam for all school graduates). Typically, these tasks are present not only in part A (the easiest test part of the exam), but also in part C (the most complex and voluminous tasks). The exam requires accurate and perfect knowledge of the topic “Natural logarithms”.

Examples and solutions to problems are taken from official Unified State Exam options. Let's see how such tasks are solved.

Given log 2 (2x-1) = 4. Solution:
let's rewrite the expression, simplifying it a little log 2 (2x-1) = 2 2, by the definition of the logarithm we get that 2x-1 = 2 4, therefore 2x = 17; x = 8.5.

• It is best to reduce all logarithms to the same base so that the solution is not cumbersome and confusing.
• All expressions under the logarithm sign are indicated as positive, therefore, when the exponent of an expression that is under the logarithm sign and as its base is taken out as a multiplier, the expression remaining under the logarithm must be positive.