What is called the logarithm of a number. Logarithmic Expressions

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log a r b r =log a b or log a b= log a r b r

The value of a logarithm will not change if the base of the logarithm and the number under the logarithm sign are raised to the same power.

Under the logarithm sign there can only be positive numbers, and the base of the logarithm is not equal to one.

Examples.

1) Compare log 3 9 and log 9 81.

log 3 9=2, since 3 2 =9;

log 9 81=2, since 9 2 =81.

So log 3 9=log 9 81.

Note that the base of the second logarithm is equal to the square of the base of the first logarithm: 9=3 2, and the number under the sign of the second logarithm is equal to the square of the number under the sign of the first logarithm: 81=9 2. It turns out that both the number and the base of the first logarithm log 3 9 were raised to the second power, and the value of the logarithm did not change:

Next, since extracting the root n th degree from among A is the raising of a number A to the degree ( 1/n), then from log 9 81 you can get log 3 9 by taking the square root of the number and the base of the logarithm:

2) Check equality: log 4 25=log 0.5 0.2.

Let's look at the first logarithm. Let's extract square root from the base 4 and from among 25 ; we get: log 4 25=log 2 5.

Let's look at the second logarithm. Logarithm base: 0.5= 1 / 2. The number under the sign of this logarithm: 0.2= 1/5. Let's raise each of these numbers to the minus first power:

0,5 -1 =(1 / 2) -1 =2;

0,2 -1 =(1 / 5) -1 =5.

So log 0.5 0.2=log 2 5. Conclusion: this equality is true.

Solve the equation:

log 4 x 4 +log 16 81=log 2 (5x+2). Let's reduce logarithms from the left to the base 2 .

log 2 x 2 +log 2 3=log 2 (5x+2). Take the square root of the number and the base of the first logarithm. Extract the fourth root of the number and the base of the second logarithm.

log 2 (3x 2)=log 2 (5x+2). Convert the sum of logarithms into the logarithm of the product.

3x 2 =5x+2. Received after potentiation.

3x 2 -5x-2=0. Let's decide quadratic equation By general formula for a complete quadratic equation:

a=3, b=-5, c=-2.

D=b 2 -4ac=(-5) 2 -4∙3∙(-2)=25+24=49=7 2 >0; 2 real roots.

Examination.

x=2.

log 4 2 4 +log 16 81=log 2 (5∙2+2);

log 2 2 2 +log 2 3=log 2 12;

log 2 (4∙3)=log 2 12;

log 2 12=log 2 12;

log a n b=(1/ n)∙ log a b

Logarithm of a number b based on a n equal to the product fractions 1/ n to the logarithm of a number b based on a.

Find:1) 21log 8 3+40log 25 2; 2) 30log 32 3∙log 125 2 , if it is known that log 2 3=b,log 5 2=c.

Solution.

Solve equations:

1) log 2 x+log 4 x+log 16 x=5.25.

Solution.

Let's reduce these logarithms to base 2. Apply the formula: log a n b=(1/ n)∙ log a b

log 2 x+(½) log 2 x+(¼) log 2 x=5.25;

log 2 x+0.5log 2 x+0.25log 2 x=5.25. Here are similar terms:

(1+0.5+0.25) log 2 x=5.25;

1.75 log 2 x=5.25 |:1.75

log 2 x=3. By definition of logarithm:

2) 0.5log 4 (x-2)+log 16 (x-3)=0.25.

Solution. Let's convert the logarithm to base 16 to base 4.

0.5log 4 (x-2)+0.5log 4 (x-3)=0.25 |:0.5

log 4 (x-2)+log 4 (x-3)=0.5. Let's convert the sum of logarithms into the logarithm of the product.

log 4 ((x-2)(x-3))=0.5;

log 4 (x 2 -2x-3x+6)=0.5;

log 4 (x 2 -5x+6)=0.5. By definition of logarithm:

x 2 -5x+4=0. According to Vieta's theorem:

x 1 =1; x 2 =4. The first value of x will not work, since at x = 1 the logarithms of this equality do not exist, because Only positive numbers can be under the logarithm sign.

Let's check this equation at x=4.

Examination.

0.5log 4 (4-2)+log 16 (4-3)=0.25

0.5log 4 2+log 16 1=0.25

0,5∙0,5+0=0,25

log a b=log c b/log c a

Logarithm of a number b based on A equal to the logarithm numbers b on a new basis With, divided by the logarithm of the old base A on a new basis With.

Examples:

1) log 2 3=lg3/lg2;

2) log 8 7=ln7/ln8.

Calculate:

1) log 5 7, if it is known that lg7≈0,8451; lg5≈0,6990.

c b / log c a.

log 5 7=log7/log5≈0.8451:0.6990≈1.2090.

Answer: log 5 7≈1,209 0≈1,209 .

2) log 5 7 , if it is known that ln7≈1,9459; ln5≈1,6094.

Solution. Apply the formula: log a b =log c b / log c a.

log 5 7=ln7/ln5≈1.9459:1.6094≈1.2091.

Answer: log 5 7≈1,209 1≈1,209 .

Find x:

1) log 3 x=log 3 4+log 5 6/log 5 3+log 7 8/log 7 3.

We use the formula: log c b / log c a = log a b . We get:

log 3 x=log 3 4+log 3 6+log 3 8;

log 3 x=log 3 (4∙6∙8);

log 3 x=log 3 192;

x=192 .

2) log 7 x=lg143-log 6 11/log 6 10-log 5 13/log 5 10.

We use the formula: log c b / log c a = log a b . We get:

log 7 x=lg143-lg11-lg13;

log 7 x=lg143- (lg11+lg13);

log 7 x=lg143-lg (11∙13);

log 7 x=lg143-lg143;

x=1.

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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 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.

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 it is necessary 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 it is necessary to raise the base “a” 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:

Natural logarithm ln a, where the base is the Euler number (e = 2.7).
Decimal a, where the base is 10.
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 an even root from 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 of an unknown degree, you need to learn how to work with a 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 large values you will need a table of degrees. 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 inequalities are defined as the region acceptable values, and the breakpoints of this function. 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.

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.
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 obligatory 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.
The logarithm of the quotient looks like this: log a (s 1/ s 2) = log a s 1 - log a s 2.
The theorem in the form of a formula takes on next view: 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 unknown value There is no such thing as a logarithm, but you can apply it 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. For solutions 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.

The property of the logarithm of a product can be used in tasks where it is necessary to expand great value numbers b into simpler factors. For example, log 2 4 + log 2 128 = log 2 (4*128) = log 2 512. The answer is 9.
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). Usually these tasks are present not only in part A (the easiest test part 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.

As society developed and production became more complex, mathematics also developed. Movement from simple to complex. From ordinary accounting using the method of addition and subtraction, with their repeated repetition, we came to the concept of multiplication and division. Reducing the repeated operation of multiplication became the concept of exponentiation. The first tables of the dependence of numbers on the base and the number of exponentiation were compiled back in the 8th century by the Indian mathematician Varasena. From them you can count the time of occurrence of logarithms.

Historical sketch

The revival of Europe in the 16th century also stimulated the development of mechanics. T required a large amount of computation related to multiplication and division of multi-digit numbers. The ancient tables were of great service. They allowed replacement complex operations to simpler ones - addition and subtraction. A big step forward was the work of the mathematician Michael Stiefel, published in 1544, in which he realized the idea of many mathematicians. This made it possible to use tables not only for degrees in the form prime numbers, but also for arbitrary rational ones.

In 1614, the Scotsman John Napier, developing these ideas, first introduced the new term “logarithm of a number.” New complex tables for calculating logarithms of sines and cosines, as well as tangents. This greatly reduced the work of astronomers.

New tables began to appear, which were successfully used by scientists for three centuries. A lot of time passed before the new operation in algebra acquired its finished form. The definition of the logarithm was given and its properties were studied.

Only in the 20th century, with the advent of the calculator and computer, did humanity abandon the ancient tables that had worked successfully throughout the 13th centuries.

Today we call the logarithm of b to base a the number x that is the power of a to make b. This is written as a formula: x = log a(b).

For example, log 3(9) would be equal to 2. This is obvious if you follow the definition. If we raise 3 to the power of 2, we get 9.

Thus, the formulated definition sets only one restriction: the numbers a and b must be real.

Types of logarithms

The classic definition is called the real logarithm and is actually the solution to the equation a x = b. Option a = 1 is borderline and is not of interest. Attention: 1 to any power is equal to 1.

Real value of logarithm defined only when the base and the argument are greater than 0, and the base must not be equal to 1.

Special place in the field of mathematics play logarithms, which will be named depending on the size of their base:

Rules and restrictions

The fundamental property of logarithms is the rule: the logarithm of a product is equal to the logarithmic sum. log abp = log a(b) + log a(p).

As a variant of this statement there will be: log c(b/p) = log c(b) - log c(p), the quotient function is equal to the difference of the functions.

From the previous two rules it is easy to see that: log a(b p) = p * log a(b).

Other properties include:

Comment. Don't make a common mistake - the logarithm of the sum is not equal to the sum logarithms.

For many centuries, the operation of finding a logarithm was a rather time-consuming task. Mathematicians used the well-known formula of the logarithmic theory of polynomial expansion:

ln (1 + x) = x — (x^2)/2 + (x^3)/3 — (x^4)/4 + … + ((-1)^(n + 1))*(( x^n)/n), where n - natural number greater than 1, which determines the accuracy of the calculation.

Logarithms with other bases were calculated using the theorem about the transition from one base to another and the property of the logarithm of the product.

Since this method is very labor-intensive and when solving practical problems difficult to implement, we used pre-compiled tables of logarithms, which significantly speeded up all the work.

In some cases, specially compiled graphs of logarithms were used, which gave less accuracy, but significantly speeded up the search for the desired value. The curve of the function y = log a(x), constructed over several points, allows you to use a regular ruler to find the value of the function at any other point. Engineers long time For these purposes, so-called graph paper was used.

In the 17th century, the first auxiliary analog computing conditions appeared, which 19th century acquired a finished look. The most successful device was called the slide rule. Despite the simplicity of the device, its appearance significantly accelerated the process of all engineering calculations, and this is difficult to overestimate. Currently, few people are familiar with this device.

The advent of calculators and computers made the use of any other devices pointless.

Equations and inequalities

To solve various equations and inequalities using logarithms, the following formulas are used:

Moving from one base to another: log a(b) = log c(b) / log c(a);
As a consequence of the previous option: log a(b) = 1 / log b(a).

To solve inequalities it is useful to know:

The value of the logarithm will be positive only if the base and argument are both greater or less than one; if at least one condition is violated, the logarithm value will be negative.
If the logarithm function is applied to the right and left sides of an inequality, and the base of the logarithm is greater than one, then the sign of the inequality is preserved; otherwise it changes.

Sample problems

Let's consider several options for using logarithms and their properties. Examples with solving equations:

Consider the option of placing the logarithm in a power:

Problem 3. Calculate 25^log 5(3). Solution: in the conditions of the problem, the entry is similar to the following (5^2)^log5(3) or 5^(2 * log 5(3)). Let's write it differently: 5^log 5(3*2), or the square of a number as a function argument can be written as the square of the function itself (5^log 5(3))^2. Using the properties of logarithms, this expression is equal to 3^2. Answer: as a result of the calculation we get 9.

Practical Application

Being a purely mathematical tool, it seems far from real life that the logarithm suddenly acquired great importance for describing objects real world. It is difficult to find a science where it is not used. This fully applies not only to natural, but also to humanitarian fields of knowledge.

Logarithmic dependencies

Here are some examples of numerical dependencies:

Mechanics and physics

Historically, mechanics and physics have always developed using mathematical methods research and at the same time served as an incentive for the development of mathematics, including logarithms. The theory of most laws of physics is written in the language of mathematics. Let us give only two examples of describing physical laws using the logarithm.

The problem of calculating such a complex quantity as the speed of a rocket can be solved by using the Tsiolkovsky formula, which laid the foundation for the theory of space exploration:

V = I * ln (M1/M2), where

V is the final speed of the aircraft.
I – specific impulse of the engine.
M 1 – initial mass of the rocket.
M 2 – final mass.

Another important example- this is used in the formula of another great scientist Max Planck, which serves to evaluate the equilibrium state in thermodynamics.

S = k * ln (Ω), where

S – thermodynamic property.
k – Boltzmann constant.
Ω is the statistical weight of different states.

Chemistry

Less obvious is the use of formulas in chemistry containing the ratio of logarithms. Let's give just two examples:

Nernst equation, the condition of the redox potential of the medium in relation to the activity of substances and the equilibrium constant.
The calculation of such constants as the autolysis index and the acidity of the solution also cannot be done without our function.

Psychology and biology

And it’s not at all clear what psychology has to do with it. It turns out that the strength of sensation is well described by this function as the inverse ratio of the stimulus intensity value to the lower intensity value.

After the above examples, it is no longer surprising that the topic of logarithms is widely used in biology. About biological forms, corresponding to logarithmic spirals, one can write entire volumes.

Other areas

It seems that the existence of the world is impossible without connection with this function, and it rules all laws. Especially when the laws of nature are associated with geometric progression. It’s worth turning to the MatProfi website, and there are many such examples in the following areas of activity: