Logarithm of a 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, except for unity, can serve as the basis of 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 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 logarithm base. 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 negative integer powers 10. The usual logarithms of the remaining numbers are enclosed between the logarithms of the nearest integer powers of 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 decimal, enclosed between 0 and 1. The integer part is 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 contained 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 the linear parts we find column 4 . At the intersection of row 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.

The tables and properties of natural logarithms are similar to the tables and properties of ordinary logarithms. The main difference between both is that the integer part natural logarithm is not significant in determining the position of the decimal point, and therefore the distinction 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; 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 published tables of the so-called in 1802. 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 by logarithms and are used in calculations when one has to deal with products and quotients. Cologarithm of a number n equal to the logarithm of the 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.

First logarithms in force 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 thicker and sparse dots) increases in arithmetic progression, When a increases in geometric progression. 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 abscissas - 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 the 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 law. 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 a quantity, such as a population, a quantity of radioactive material, or a bank balance, decreases or increases at a rate proportional to the available at the 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, mountain sheep horns 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 occurs in such distant and completely different areas as the contour of an eccentric cam and the trajectory of some insects flying towards the light.

often take a number e = 2,718281828 . Logarithms based on this base are called natural. When performing calculations with natural logarithms, it is common to operate with the sign ln, not log; while the number 2,718281828 , defining the basis, are not indicated.

In other words, the formulation will look like: natural logarithm numbers X- this is an exponent to which a number must be raised e to get x.

So, ln(7,389...)= 2, since e 2 =7,389... . Natural logarithm of the number itself e= 1 because e 1 =e, and the natural logarithm of unity is zero, since e 0 = 1.

The number itself e defines the limit of a monotone bounded sequence

calculated that e = 2,7182818284... .

Quite often, in order to fix a number in memory, the digits of the required number are associated with some outstanding date. Speed ​​of memorizing the first nine digits of a number e after the decimal point will increase if you notice that 1828 is the year of birth of Leo Tolstoy!

Today there are enough full tables natural logarithms.

Natural logarithm graph(functions y=ln x) is a consequence of the exponential graph mirror image relatively straight y = x and has the form:

The natural logarithm can be found for every positive real number a as the area under the curve y = 1/x from 1 to a.

The elementary nature of this formulation, which is consistent with many other formulas in which the natural logarithm is involved, was the reason for the formation of the name “natural”.

If you analyze natural logarithm, as a real function of a real variable, then it acts inverse function to an exponential function, which reduces to the identities:

e ln(a) =a (a>0)

ln(e a) =a

By analogy with all logarithms, the natural logarithm converts multiplication into addition, division into subtraction:

ln(xy) = ln(x) + ln(y)

ln(x/y)= lnx - lny

The logarithm can be found for every positive base that is not equal to one, not just for 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.

Having analyzed natural logarithm graph, we find that it exists for positive values variable x. It increases monotonically in its domain of definition.

At x → 0 the limit of the natural logarithm is minus infinity ( -∞ ).At x → +∞ the limit of the natural logarithm is plus infinity ( + ∞ ). At large x The logarithm increases quite slowly. Any power function xa with a positive exponent a increases faster than the logarithm. The natural logarithm is a monotonically increasing function, so it has no extrema.

Usage natural logarithms very rational when passing higher mathematics. Thus, using the logarithm is convenient for finding the answer to equations in which unknowns appear as exponents. The use of natural logarithms in calculations makes it possible to greatly simplify large number mathematical formulas. Logarithms to the base e are present in solving a significant number of physical problems and are naturally included in the mathematical description of individual chemical, biological and other processes. Thus, logarithms are used to calculate the decay constant for a known half-life, or to calculate the decay time in solving problems of radioactivity. They perform in leading role in many branches of mathematics and practical sciences, they are resorted to in the field of finance to solve large number tasks, including the calculation of compound interest.

Instructions

Write the given logarithmic expression. If the expression uses the logarithm of 10, then its notation is shortened and looks like this: lg b is the decimal logarithm. If the logarithm has the number e as its base, then write the expression: ln b – natural logarithm. It is understood that the result of any is the power to which the base number must be raised to obtain the number b.

When finding the sum of two functions, you simply need to differentiate them one by one and add the results: (u+v)" = u"+v";

When finding the derivative of the product of two functions, it is necessary to multiply the derivative of the first function by the second and add the derivative of the second function multiplied by the first function: (u*v)" = u"*v+v"*u;

In order to find the derivative of the quotient of two functions, it is necessary to subtract from the product of the derivative of the dividend multiplied by the divisor function the product of the derivative of the divisor multiplied by the function of the dividend, and divide all this by the divisor function squared. (u/v)" = (u"*v-v"*u)/v^2;

If given complex function, then it is necessary to multiply the derivative of the internal function and the derivative of the external one. Let y=u(v(x)), then y"(x)=y"(u)*v"(x).

Using the results obtained above, you can differentiate almost any function. So let's look at a few examples:

y=x^4, y"=4*x^(4-1)=4*x^3;

y=2*x^3*(e^x-x^2+6), y"=2*(3*x^2*(e^x-x^2+6)+x^3*(e^x-2 *x));
There are also problems involving calculating the derivative at a point. Let the function y=e^(x^2+6x+5) be given, you need to find the value of the function at the point x=1.
1) Find the derivative of the function: y"=e^(x^2-6x+5)*(2*x +6).

2) Calculate the value of the function at a given point y"(1)=8*e^0=8

Video on the topic

Useful advice

Learn the table of elementary derivatives. This will significantly save time.

Sources:

• derivative of a constant

So, what is the difference between an irrational equation and a rational one? If the unknown variable is under the sign square root, then the equation is considered irrational.

Instructions

The main method for solving such equations is the method of constructing both sides equations into a square. However. this is natural, the first thing you need to do is get rid of the sign. This method is not technically difficult, but sometimes it can lead to trouble. For example, the equation is v(2x-5)=v(4x-7). By squaring both sides you get 2x-5=4x-7. Solving such an equation is not difficult; x=1. But the number 1 will not be given equations. Why? Substitute one into the equation instead of the value of x. And the right and left sides will contain expressions that do not make sense, that is. This value is not valid for a square root. Therefore, 1 is an extraneous root, and therefore this equation has no roots.

So, an irrational equation is solved using the method of squaring both its sides. And having solved the equation, it is necessary to cut off extraneous roots. To do this, substitute the found roots into the original equation.

Consider another one.
2х+vх-3=0
Of course, this equation can be solved using the same equation as the previous one. Move Compounds equations, which do not have a square root, to the right side and then use the squaring method. solve the resulting rational equation and roots. But also another, more elegant one. Enter a new variable; vх=y. Accordingly, you will receive an equation of the form 2y2+y-3=0. That is, the usual quadratic equation. Find its roots; y1=1 and y2=-3/2. Next, solve two equations vх=1; vх=-3/2. The second equation has no roots; from the first we find that x=1. Don't forget to check the roots.

Solving identities is quite simple. To do this, it is necessary to carry out identical transformations until the set goal is achieved. Thus, with the help of simple arithmetic operations, the problem posed will be solved.

You will need

• - paper;
• - pen.

Instructions

The simplest of such transformations are algebraic abbreviated multiplications (such as the square of the sum (difference), difference of squares, sum (difference), cube of the sum (difference)). In addition, there are many trigonometric formulas, which are essentially the same identities.

Indeed, the square of the sum of two terms is equal to the square of the first plus twice the product of the first by the second and plus the square of the second, that is (a+b)^2= (a+b)(a+b)=a^2+ab +ba+b ^2=a^2+2ab+b^2.

Simplify both

General principles of the solution

Repeat from a textbook on mathematical analysis or higher mathematics what a definite integral is. As is known, the solution to a definite integral is a function whose derivative will give an integrand. This function is called antiderivative. Based on this principle, the basic integrals are constructed.
Determine by the form of the integrand which of the table integrals fits in in this case. It is not always possible to determine this immediately. Often, the tabular form becomes noticeable only after several transformations to simplify the integrand.

Variable Replacement Method

If the integrand function is trigonometric function, whose argument contains some polynomial, then try using the variable replacement method. In order to do this, replace the polynomial in the argument of the integrand with some new variable. Based on the relationship between the new and old variables, determine the new limits of integration. By differentiating this expression, find the new differential in . So you will get new look of the previous integral, close to or even corresponding to any tabular one.

Solving integrals of the second kind

If the integral is an integral of the second kind, a vector form of the integrand, then you will need to use the rules for the transition from these integrals to scalar ones. One such rule is the Ostrogradsky-Gauss relation. This law allows us to move from the rotor flux of a certain vector function to the triple integral over the divergence of a given vector field.

Substitution of integration limits

After finding the antiderivative, it is necessary to substitute the limits of integration. First, substitute the value of the upper limit into the expression for the antiderivative. You will get some number. Next, subtract from the resulting number another number obtained from the lower limit into the antiderivative. If one of the limits of integration is infinity, then when substituting it into the antiderivative function, it is necessary to go to the limit and find what the expression tends to.
If the integral is two-dimensional or three-dimensional, then you will have to represent the limits of integration geometrically to understand how to evaluate the integral. Indeed, in the case of, say, a three-dimensional integral, the limits of integration can be entire planes that limit the volume being integrated.

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 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 individual species 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 decided in a standard way, which includes simplification, reduction and subsequent reduction to one logarithm using logarithmic theorems. To receive correct values 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 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 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 numerical values ​​in the answer, while when solving an inequality, both the range of acceptable values ​​​​and the points are determined breaking 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.

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 prerequisite 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. For admission to university or passing entrance examinations in mathematics you need to know how to solve such problems correctly.

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. Simplify long ones logarithmic expressions possible 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 of 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 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.
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 many 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.

1.1. Determining the exponent for an integer exponent

X 1 = X
X 2 = X * X
X 3 = X * X * X
…
X N = X * X * … * X - N times

1.2. Zero degree.

By definition, it is generally accepted that the zero power of any number is 1:

1.3. Negative degree.

X -N = 1/X N

1.4. Fractional power, root.

X 1/N = N root of X.

For example: X 1/2 = √X.

1.5. Formula for adding powers.

X (N+M) = X N *X M

1.6.Formula for subtracting powers.

X (N-M) = X N /X M

1.7. Formula for multiplying powers.

X N*M = (X N) M

1.8. Formula for raising a fraction to a power.

(X/Y) N = X N /Y N

2. Number e.

The value of the number e is equal to the following limit:

E = lim(1+1/N), as N → ∞.

With an accuracy of 17 digits, the number e is 2.71828182845904512.

3. Euler's equality.

This equality connects five numbers that play a special role in mathematics: 0, 1, e, pi, imaginary unit.

E (i*pi) + 1 = 0

4. Exponential function exp(x)

exp(x) = e x

5. Derivative of exponential function

The exponential function has remarkable property: The derivative of a function is equal to the exponential function itself:

(exp(x))" = exp(x)

6. Logarithm.

6.1. Definition of the logarithm function

If x = b y, then the logarithm is the function

Y = Log b(x).

The logarithm shows to what power a number - the base of the logarithm (b) - must be raised to obtain a given number (X). The logarithm function is defined for X greater than zero.

For example: Log 10 (100) = 2.

6.2. Decimal logarithm

This is the logarithm to base 10:

Y = Log 10 (x) .

Denoted by Log(x): Log(x) = Log 10 (x).

Usage example decimal logarithm- decibel.

6.3. Decibel

The item is highlighted on a separate page Decibel

6.4. Binary logarithm

This is the base 2 logarithm:

Y = Log 2 (x).

Denoted by Lg(x): Lg(x) = Log 2 (X)

6.5. Natural logarithm

This is the logarithm to base e:

Y = Log e (x) .

Denoted by Ln(x): Ln(x) = Log e (X)
The natural logarithm is the inverse function of the exponential function exp(X).

6.6. Characteristic points

Loga(1) = 0
Log a (a) = 1

6.7. Product logarithm formula

Log a (x*y) = Log a (x)+Log a (y)

6.8. Formula for logarithm of quotient

Log a (x/y) = Log a (x)-Log a (y)

6.9. Logarithm of power formula

Log a (x y) = y*Log a (x)

6.10. Formula for converting to a logarithm with a different base

Log b (x) = (Log a (x))/Log a (b)

Example:

Log 2 (8) = Log 10 (8)/Log 10 (2) =
0.903089986991943552 / 0.301029995663981184 = 3

7. Formulas useful in life

Often there are problems of converting volume into area or length and the inverse problem - converting area into volume. For example, boards are sold in cubes (cubic meters), and we need to calculate how much wall area can be covered with boards contained in a certain volume, see calculation of boards, how many boards are in a cube. Or, if the dimensions of the wall are known, you need to calculate the number of bricks, see brick calculation.

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