• Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the Beginnings of Analysis: A Textbook for Grades 10-11 of General Educational Institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).

With the development of society, the complexity of production, mathematics also developed. Movement from simple to complex. From the usual accounting method of addition and subtraction, with their repeated repetition, they came to the concept of multiplication and division. The reduction of the multiply repeated operation 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 outline

The revival of Europe in the 16th century also stimulated the development of mechanics. T required a large amount of computation associated with multiplication and division of multi-digit numbers. The ancient tables did a great service. They made it possible to replace complex operations with 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 of 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 were compiled for calculating the 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 logarithm was defined and its properties were studied.

Only in the 20th century, with the advent of the calculator and the computer, mankind abandoned the ancient tables that had been successfully operating throughout the 13th centuries.

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

For example, log 3(9) will 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 puts only one restriction, the numbers a and b must be real.

Varieties of logarithms

The classical definition is called the real logarithm and is actually a solution to the equation a x = b. The option a = 1 is borderline and is of no interest. Note: 1 to any power is 1.

Real value of the logarithm defined only if the base and the argument is 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 value 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, it will be: log c (b / p) \u003d log c (b) - log c (p), the quotient function is equal to the difference of the functions.

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

Other properties include:

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

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

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

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

Since this method is very laborious and when solving practical problems difficult to implement, they used pre-compiled tables of logarithms, which greatly accelerated the entire 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), built on several points, allows using the usual ruler to find the values ​​of the function at any other point. For a long time, engineers used the so-called graph paper for these purposes.

In the 17th century, the first auxiliary analog computing conditions appeared, which by the 19th century had acquired a finished form. 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 it pointless to use any other devices.

Equations and inequalities

The following formulas are used to solve various equations and inequalities using logarithms:

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

To solve inequalities, it is useful to know:

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

Task examples

Consider several options for using logarithms and their properties. Examples with solving equations:

Consider the option of placing the logarithm in the degree:

• Task 3. Calculate 25^log 5(3). Solution: in the conditions of the problem, the notation 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 3^2. Answer: as a result of the calculation we get 9.

Practical use

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

Logarithmic dependencies

Here are some examples of numerical dependencies:

Mechanics and physics

Historically, mechanics and physics have always developed using mathematical research methods 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. We give only two examples of the description of physical laws using the logarithm.

It is possible to solve the problem of calculating such a complex quantity as the speed of a rocket 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 is the specific impulse of the engine.
• M 1 is the initial mass of the rocket.
• M 2 - final mass.

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

S = k * ln (Ω), where

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

Chemistry

Less obvious would be the use of formulas in chemistry containing the ratio of logarithms. Here are just two examples:

• The 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 autoprolysis index and the acidity of the solution is also not complete without our function.

Psychology and biology

And it’s completely incomprehensible what the 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 theme of logarithms is also widely used in biology. Whole volumes can be written about biological forms corresponding to logarithmic spirals.

Other areas

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

The list could be endless. Having mastered the basic laws of this function, you can plunge into the world of infinite wisdom.

What is a logarithm?

Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")

What is a logarithm? How to solve logarithms? These questions confuse many graduates. Traditionally, the topic of logarithms is considered complex, incomprehensible and scary. Especially - equations with logarithms.

This is absolutely not true. Absolutely! Don't believe? Good. Now, for some 10 - 20 minutes you:

1. Understand what is a logarithm.

2. Learn to solve a whole class of exponential equations. Even if you haven't heard of them.

3. Learn to calculate simple logarithms.

Moreover, for this you will only need to know the multiplication table, and how a number is raised to a power ...

I feel you doubt ... Well, keep time! Go!

First, solve the following equation in your mind:

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

In relation to

the task of finding any of the three numbers from the other two, given, can be set. Given a and then N is found by exponentiation. If N are given and then a is found by extracting the root of the power x (or exponentiation). Now consider the case when, given a and N, it is required to find x.

Let the number N be positive: the number a is positive and not equal to one: .

Definition. The logarithm of the number N to the base a is the exponent to which you need to raise a to get the number N; the logarithm is denoted by

Thus, in equality (26.1), the exponent is found as the logarithm of N to the base a. Entries

have the same meaning. Equality (26.1) is sometimes called the basic identity of the theory of logarithms; in fact, it expresses the definition of the concept of the logarithm. By this definition, the base of the logarithm a is always positive and different from unity; the logarithmable number N is positive. Negative numbers and zero do not have logarithms. It can be proved that any number with a given base has a well-defined logarithm. Therefore equality entails . Note that the condition is essential here, otherwise the conclusion would not be justified, since the equality is true for any values ​​of x and y.

Example 1. Find

Solution. To get the number, you need to raise base 2 to the power Therefore.

You can record when solving such examples in the following form:

Example 2. Find .

Solution. We have

In examples 1 and 2, we easily found the desired logarithm by representing the logarithmable number as a degree of base with a rational exponent. In the general case, for example, for etc., this cannot be done, since the logarithm has an irrational value. Let us pay attention to one question related to this statement. In § 12 we gave the concept of the possibility of determining any real power of a given positive number. This was necessary for the introduction of logarithms, which, in general, can be irrational numbers.

Consider some properties of logarithms.

Property 1. If the number and base are equal, then the logarithm is equal to one, and, conversely, if the logarithm is equal to one, then the number and base are equal.

Proof. Let By the definition of the logarithm, we have and whence

Conversely, let Then by definition

Property 2. The logarithm of unity to any base is equal to zero.

Proof. By the definition of the logarithm (the zero power of any positive base is equal to one, see (10.1)). From here

Q.E.D.

The converse statement is also true: if , then N = 1. Indeed, we have .

Before stating the following property of logarithms, let us agree to say that two numbers a and b lie on the same side of a third number c if they are both either greater than c or less than c. If one of these numbers is greater than c and the other is less than c, then we say that they lie on opposite sides of c.

Property 3. If the number and base lie on the same side of unity, then the logarithm is positive; if the number and base lie on opposite sides of unity, then the logarithm is negative.

The proof of property 3 is based on the fact that the degree of a is greater than one if the base is greater than one and the exponent is positive, or the base is less than one and the exponent is negative. The degree is less than one if the base is greater than one and the exponent is negative, or the base is less than one and the exponent is positive.

There are four cases to be considered:

We confine ourselves to the analysis of the first of them, the reader will consider the rest on his own.

Let then the exponent in equality be neither negative nor equal to zero, therefore, it is positive, i.e., which was required to be proved.

Example 3. Find out which of the following logarithms are positive and which are negative:

Solution, a) since the number 15 and the base 12 are located on the same side of the unit;

b) , since 1000 and 2 are located on the same side of the unit; at the same time, it is not essential that the base is greater than the logarithmic number;

c), since 3.1 and 0.8 lie on opposite sides of unity;

G) ; why?

e) ; why?

The following properties 4-6 are often called the rules of logarithm: they allow, knowing the logarithms of some numbers, to find the logarithms of their product, quotient, degree of each of them.

Property 4 (the rule for the logarithm of the product). The logarithm of the product of several positive numbers in a given base is equal to the sum of the logarithms of these numbers in the same base.

Proof. Let positive numbers be given.

For the logarithm of their product, we write the equality (26.1) defining the logarithm:

From here we find

Comparing the exponents of the first and last expressions, we obtain the required equality:

Note that the condition is essential; the logarithm of the product of two negative numbers makes sense, but in this case we get

In general, if the product of several factors is positive, then its logarithm is equal to the sum of the logarithms of the modules of these factors.

Property 5 (quotient logarithm rule). The logarithm of a quotient of positive numbers is equal to the difference between the logarithms of the dividend and the divisor, taken in the same base. Proof. Consistently find

Q.E.D.

Property 6 (rule of the logarithm of the degree). The logarithm of the power of any positive number is equal to the logarithm of that number times the exponent.

Proof. We write again the main identity (26.1) for the number :

Q.E.D.

Consequence. The logarithm of the root of a positive number is equal to the logarithm of the root number divided by the exponent of the root:

We can prove the validity of this corollary by presenting how and using property 6.

Example 4. Logarithm to base a:

a) (it is assumed that all values ​​b, c, d, e are positive);

b) (it is assumed that ).

Solution, a) It is convenient to pass in this expression to fractional powers:

Based on equalities (26.5)-(26.7) we can now write:

We notice that simpler operations are performed on the logarithms of numbers than on the numbers themselves: when multiplying numbers, their logarithms are added, when divided, they are subtracted, etc.

That is why logarithms have been used in computational practice (see Sec. 29).

The action inverse to the logarithm is called potentiation, namely: potentiation is the action by which this number itself is found by the given logarithm of a number. In essence, potentiation is not any special action: it comes down to raising the base to a power (equal to the logarithm of the number). The term "potentiation" can be considered synonymous with the term "exponentiation".

When potentiating, it is necessary to use the rules that are inverse to the rules of logarithm: replace the sum of logarithms with the logarithm of the product, the difference of logarithms with the logarithm of the quotient, etc. In particular, if there is any factor in front of the sign of the logarithm, then during potentiation it must be transferred to the indicator degrees under the sign of the logarithm.

Example 5. Find N if it is known that

Solution. In connection with the potentiation rule just stated, the factors 2/3 and 1/3, which are in front of the signs of logarithms on the right side of this equality, will be transferred to the exponents under the signs of these logarithms; we get

Now we replace the difference of logarithms with the logarithm of the quotient:

to obtain the last fraction in this chain of equalities, we freed the previous fraction from irrationality in the denominator (section 25).

Property 7. If the base is greater than one, then the larger number has a larger logarithm (and the smaller one has a smaller one), if the base is less than one, then the larger number has a smaller logarithm (and the smaller one has a larger one).

This property is also formulated as a rule for the logarithm of inequalities, both parts of which are positive:

When taking the logarithm of inequalities to a base greater than one, the inequality sign is preserved, and when taking a logarithm to a base less than one, the sign of the inequality is reversed (see also item 80).

The proof is based on properties 5 and 3. Consider the case when If , then and, taking the logarithm, we obtain

(a and N/M lie on the same side of unity). From here

Case a follows, the reader will figure it out for himself.