What does ln stand for. Natural logarithm and the number e. The formula for subtracting powers

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) & nbsp & nbsp & nbsp & nbsp & nbsp & nbsp

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

Basic logarithmic identity

a log a b = b (a> 0, a ≠ 1) (2)

It is important that the domains of definition of the right and left sides of this formula are different. The left-hand 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 use of the basic logarithmic "identity" in solving equations and inequalities can lead to a change in the GDV.

Two obvious consequences of the definition of a logarithm

log a a = 1 (a> 0, a ≠ 1) (3)
log a 1 = 0 (a> 0, a ≠ 1) (4)

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 the logarithm of the quotient

log a (b c) = log a b + log a c (a> 0, a ≠ 1, b> 0, c> 0) (5)

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 thoughtless use of these formulas when solving logarithmic equations and inequalities. When they are used "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 ODV 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 have to limit ourselves only to the case when f (x)> 0 and g (x)> 0. There is a narrowing of the range of permissible values, and this is categorically unacceptable, since it can lead to the loss of solutions. A similar problem exists for formula (6).

The degree can be expressed outside the sign of the logarithm

log a b p = p log a b (a> 0, a ≠ 1, b> 0) (7)

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-hand side of the equality is defined, obviously, for all values ​​of f (x), except zero. The right side is only for f (x)> 0! Taking the degree out of the logarithm, we again narrow the ODV. The reverse procedure expands the range of valid values. All these remarks apply not only to degree 2, but also to any even degree.

The formula for the transition to a new base

log a b = log c b log c a (a> 0, a ≠ 1, b> 0, c> 0, c ≠ 1) (8)

This is the rare case when the ODV does not change during the transformation. If you have reasonably chosen a base c (positive and not equal to 1), the formula for transition to a new base is completely safe.

If we choose the number b as the new base c, we get an important special case of formula (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: lg2 + lg50.
Solution. lg2 + lg50 = lg100 = 2. We used the formula for the sum of logarithms (5) and the definition of the decimal logarithm.

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

Table of formulas related to logarithms

a log a b = b (a> 0, a ≠ 1)

log a a = 1 (a> 0, a ≠ 1)

log a 1 = 0 (a> 0, a ≠ 1)

log a (b c) = log a b + log a c (a> 0, a ≠ 1, b> 0, c> 0)

log a b c = log a b - log a c (a> 0, a ≠ 1, b> 0, c> 0)

log a b p = p log a b (a> 0, a ≠ 1, b> 0)

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

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

By the base of the number e: ln x = log e x.

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

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

Function graph y = ln x.

Natural logarithm graph (functions y = ln x) is obtained from the exponent graph by mirroring it relative to the straight line y = x.

The natural logarithm is defined for positive values ​​of the variable x. It increases monotonically on its domain of definition.

As 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 rather slowly. Any power function x a with a positive exponent a grows faster than the logarithm.

Natural logarithm properties

Range of definition, set of values, extrema, increasing, decreasing

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

Ln x

ln 1 = 0

Basic formulas for natural logarithms

Formulas arising 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 change formula:

The proofs of these formulas are presented in the "Logarithm" section.

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 the modulus x:
.
Derivative of the nth order:
.
Derivation of formulas>>>

Integral

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

Expressions in terms of complex numbers

Consider a function of a complex variable z:
.
Let us express the complex variable z via module r and the argument φ :
.
Using the properties of the logarithm, we have:
.
Or
.
The φ argument is not uniquely defined. If we 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 an unambiguous function.

Power series expansion

At the decomposition takes place:

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Technical Institutions, "Lan", 2009.

Before we get acquainted with the concept of the natural logarithm, consider the concept of a constant number $ e $.

Number $ e $

Definition 1

Number $ e $ Is a mathematical constant, which is a transcendental number and is equal to $ e \ approx 2.718281828459045 \ ldots $.

Definition 2

Transcendental is a number that is not a root of a polynomial with integer coefficients.

Remark 1

The last formula describes second wonderful limit.

The number e is also called Euler numbers and sometimes Napier numbers.

Remark 2

To remember the first signs of the number $ e $, the following expression is often used: "$ 2 $, $ 7 $, twice Leo Tolstoy"... Of course, in order to be able to use it, you must remember that Leo Tolstoy was born in $ 1828 $. It is these numbers that are repeated twice in the value of the number $ e $ after the integer part of $ 2 $ and the decimal $ 7 $.

We started considering the concept of the number $ e $ in the study of the natural logarithm precisely because it stands at the base of the logarithm $ \ log_ (e) ⁡a $, which is usually called natural and written in the form $ \ ln ⁡a $.

Natural logarithm

Often, when calculating, logarithms are used, at the base of which is the number $ e $.

Definition 4

The logarithm with base $ e $ is called natural.

Those. the natural logarithm can be denoted as $ \ log_ (e) ⁡a $, but in mathematics it is customary to use the notation $ \ ln ⁡a $.

Natural logarithm properties

Because the logarithm to any base of one is $ 0 $, then the natural logarithm of one is $ 0 $:

The natural logarithm of $ e $ is equal to one:

The natural logarithm of the product of two numbers is equal to the sum of the natural logarithms of these numbers:

$ \ ln ⁡ (ab) = \ ln ⁡a + \ ln ⁡b $.

The natural logarithm of the quotient of two numbers is equal to the difference between the natural logarithms of these numbers:

$ \ ln⁡ \ frac (a) (b) = \ ln ⁡a- \ ln⁡ b $.

The natural logarithm of the power of a number can be represented as the product of the exponent by the natural logarithm of the sub-logarithmic number:

$ \ ln⁡ a ^ s = s \ cdot \ ln⁡ a $.

Example 1

Simplify the expression $ \ frac (2 \ ln ⁡4e- \ ln ⁡16) (\ ln ⁡5e- \ frac (1) (2) \ ln ⁡25) $.

Solution.

We apply the property of the logarithm of the product to the first logarithm in the numerator and in the denominator, and the property of the logarithm of the degree to the second logarithm of the numerator and denominator:

$ \ frac (2 \ ln ⁡4e- \ ln⁡16) (\ ln ⁡5e- \ frac (1) (2) \ ln ⁡25) = \ frac (2 (\ ln ⁡4 + \ ln ⁡e) - \ ln⁡ 4 ^ 2) (\ ln ⁡5 + \ ln ⁡e- \ frac (1) (2) \ ln⁡ 5 ^ 2) = $

open the brackets and present similar terms, and also apply the property $ \ ln ⁡e = 1 $:

$ = \ frac (2 \ ln ⁡4 + 2-2 \ ln ⁡4) (\ ln ⁡5 + 1- \ frac (1) (2) \ cdot 2 \ ln ⁡5) = \ frac (2) ( \ ln ⁡5 + 1- \ ln ⁡5) = 2 $.

Answer: $ \ frac (2 \ ln ⁡4e- \ ln ⁡16) (\ ln ⁡5e- \ frac (1) (2) \ ln ⁡25) = 2 $.

Example 2

Find the value of the expression $ \ ln⁡ 2e ^ 2 + \ ln \ frac (1) (2e) $.

Solution.

Let's apply the formula for the sum of logarithms:

$ \ ln 2e ^ 2 + \ ln \ frac (1) (2e) = \ ln 2e ^ 2 \ cdot \ frac (1) (2e) = \ ln ⁡e = 1 $.

Answer: $ \ ln 2e ^ 2 + \ ln \ frac (1) (2e) = 1 $.

Example 3

Evaluate the value of the logarithmic expression $ 2 \ lg ⁡0,1 + 3 \ ln⁡ e ^ 5 $.

Solution.

Let's apply the property of the logarithm of the degree:

$ 2 \ lg ⁡0,1 + 3 \ ln e ^ 5 = 2 \ lg 10 ^ (- 1) +3 \ cdot 5 \ ln ⁡e = -2 \ lg ⁡10 + 15 \ ln ⁡e = -2 + 15 = 13 $.

Answer: $ 2 \ lg ⁡0,1 + 3 \ ln e ^ 5 = 13 $.

Example 4

Simplify the logarithmic expression for $ \ ln \ frac (1) (8) -3 \ ln ⁡4 $.

$ 3 \ ln \ frac (9) (e ^ 2) -2 \ ln ⁡27 = 3 \ ln (\ frac (3) (e)) ^ 2-2 \ ln 3 ^ 3 = 3 \ cdot 2 \ ln \ frac (3) (e) -2 \ cdot 3 \ ln ⁡3 = 6 \ ln \ frac (3) (e) -6 \ ln ⁡3 = $

we apply to the first logarithm the property of the logarithm of the quotient:

$ = 6 (\ ln ⁡3- \ ln ⁡e) -6 \ ln⁡ 3 = $

let's open the brackets and give similar terms:

$ = 6 \ ln ⁡3-6 \ ln ⁡e-6 \ ln ⁡3 = -6 $.

Answer: $ 3 \ ln \ frac (9) (e ^ 2) -2 \ ln ⁡27 = -6 $.

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

If, then.

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

In contact with

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

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

Here are some identities:

Here are the main algebraic expressions:

;

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

Let's try to understand the question of what natural logarithms are. Separate interest in mathematics are of two types- the first is based on the number "10", and is called "decimal logarithm". The second is called natural. The base of the natural logarithm is the number "e". It is about him that we will talk in detail in this article.

Legend:

• lg x - decimal;
• ln x - natural.

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

Natural logarithm plot

Let's construct a graph of the natural logarithm in the standard classical way by points. If you wish, you can check if we are building the function correctly by examining the function. However, it makes sense to learn how to build it "by hand" in order to know how to correctly calculate the logarithm.

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

Let us explain why we have chosen exactly such values ​​of the argument x. It's all about the identity:. For the natural logarithm, this identity will look like this:

For convenience, we can take five anchor points:

;

;

.

;

.

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

Having built a graph by points, we get an approximate graph:

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

Attention! Only positive numbers are included in the domain of the natural logarithm! The domain of definition does not include x = 0. This is impossible on the basis of the conditions for the existence of the logarithm.

Range of values ​​(that is, all valid values ​​of the function y = ln x) - 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 cannot do this, since the natural logarithm at x<0 не существует.

Natural limit log can be written like this:

The formula for replacing the base of a logarithm

The natural logarithm is much easier to deal with than the arbitrary base logarithm. That is why we will try to learn how to reduce any logarithm to natural, or express it in 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 logarithmized on both sides. Let's do this using an arbitrary base z:

Let's use the property (only instead of "c" we have an expression):

From here we get a universal formula:

.

In particular, if z = e, then:

.

We managed to represent the logarithm to an arbitrary base through the ratio of two natural logarithms.

We solve problems

In order to better navigate in natural logarithms, consider 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:

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

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

.

Let's apply the definition of a logarithm again:

.

Thus:

.

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

Objective 3. Solve the equation.

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

.

Before us is a quadratic equation. Let's find its discriminant:

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

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

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

In mechanics and physics there is no section where logarithms have not been used. The barometric distribution, all the principles of statistical thermodynamics, the Tsiolkovsky equation and so on are processes that can only be mathematically described using logarithms.

In chemistry, the logarithm is used in the Nernst equations, descriptions of redox processes.

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

Natural logarithm Function y = ln x its properties

Proof of the main property of the natural logarithm