The logarithm of the number b to the base a is the exponent to which you need to raise the number a to get the number b.

If , then .

The logarithm is extremely important mathematical value , since the logarithmic calculus allows not only to solve exponential equations, but also 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 the properties of 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:

;

.

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

Let's try to understand the question of what natural logarithms are. Separate interest in mathematics represent two types- the first has the number "10" at the base, 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.

Designations:

• lg x - decimal;
• ln x - natural.

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

natural log graph

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

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

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

For convenience, we can take five reference points:

;

;

.

;

.

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

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

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

Attention! The domain of the natural logarithm includes only positive numbers! The scope does not include x=0. This is impossible based on the conditions for the existence of the logarithm.

The range of values ​​(i.e. all valid values ​​of the function y = ln x) is all numbers in the interval .

natural log limit

Studying the graph, the question arises - how does the function behave when y<0.

Obviously, the graph of the function tends to cross the y-axis, but will not be able to do this, since the natural logarithm of x<0 не существует.

Natural limit log can be written like this:

Formula for changing the base of a logarithm

Dealing with a natural logarithm is much easier than dealing with a logarithm that has an arbitrary base. That is why we will try to learn how to reduce any logarithm to a natural one, 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 with an arbitrary base z:

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

From here we get the 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.

Task 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:

.

Once again, we apply the definition of the logarithm:

.

Thus:

.

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

Task 3. Solve the equation.

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

.

We have a quadratic equation. Let's find its discriminant:

First root of the equation:

.

Second root of the equation:

.

Remembering that we made the substitution t = ln x, we get:

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 bits in memory.

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

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

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

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

Natural logarithm Function y=ln x its properties

Proof of the main property of the natural logarithm

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 related to 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 from real life that the logarithm has suddenly acquired great importance to describe 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.

Today we will talk about logarithm formulas and give demonstration solution examples.

By themselves, they imply solution patterns according to the basic properties of logarithms. Before applying the logarithm formulas to the solution, we recall for you, first all the properties:

Now, based on these formulas (properties), we show examples of solving logarithms.

Examples of solving logarithms based on formulas.

Logarithm a positive number b in base a (denoted log a b) is the exponent to which a must be raised to get b, with b > 0, a > 0, and 1.

According to the definition log a b = x, which is equivalent to a x = b, so log a a x = x.

Logarithms, examples:

log 2 8 = 3, because 2 3 = 8

log 7 49 = 2 because 7 2 = 49

log 5 1/5 = -1, because 5 -1 = 1/5

Decimal logarithm is an ordinary logarithm, the base of which is 10. Denoted as lg.

log 10 100 = 2 because 10 2 = 100

natural logarithm- also the usual logarithm logarithm, but with the base e (e \u003d 2.71828 ... - an irrational number). Referred to as ln.

It is desirable to remember the formulas or properties of logarithms, because we will need them later when solving logarithms, logarithmic equations and inequalities. Let's work through each formula again with examples.

• Basic logarithmic identity
  a log a b = b

  8 2log 8 3 = (8 2log 8 3) 2 = 3 2 = 9

• The logarithm of the product is equal to the sum of the logarithms
  log a (bc) = log a b + log a c

  log 3 8.1 + log 3 10 = log 3 (8.1*10) = log 3 81 = 4

• The logarithm of the quotient is equal to the difference of the logarithms
  log a (b/c) = log a b - log a c

  9 log 5 50 /9 log 5 2 = 9 log 5 50- log 5 2 = 9 log 5 25 = 9 2 = 81

• Properties of the degree of a logarithmable number and the base of the logarithm

  The exponent of a logarithm number log a b m = mlog a b

  Exponent of the base of the logarithm log a n b =1/n*log a b

  log a n b m = m/n*log a b,

  if m = n, we get log a n b n = log a b

  log 4 9 = log 2 2 3 2 = log 2 3

• Transition to a new foundation
  log a b = log c b / log c a,

  if c = b, we get log b b = 1

  then log a b = 1/log b a

  log 0.8 3*log 3 1.25 = log 0.8 3*log 0.8 1.25/log 0.8 3 = log 0.8 1.25 = log 4/5 5/4 = -1

As you can see, the logarithm formulas are not as complicated as they seem. Now, having considered examples of solving logarithms, we can move on to logarithmic equations. We will consider examples of solving logarithmic equations in more detail in the article: "". Do not miss!

If you still have questions about the solution, write them in the comments to the article.

Note: decided to get an education of another class study abroad as an option.

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 indicators. It was they who served for the further discovery of logarithms. Examples of using this function can be found almost everywhere where it is required to simplify cumbersome multiplication to simple addition. If you spend 10 minutes reading this article, we will explain to you what logarithms are and how to work with them. Simple and accessible language.

Definition in mathematics

The 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" according to its base "a" is considered the power of "c", to which it is necessary to raise the base "a", so that in the end 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 such a degree that from 2 to the required degree you get 8. Having done some calculations in your mind, we get the number 3! And rightly so, because 2 to the power of 3 gives the number 8 in the answer.

Varieties 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 certain types 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. The logarithm of any number b to the base a>1.

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

Rules and some restrictions

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

• the base "a" must always be greater than zero, and at the same time not be 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 be greater than zero.

How to solve logarithms?

For example, the task was given to find the answer to the equation 10 x \u003d 100. It is very easy, you need to choose such a power, raising the number ten to which we get 100. This, of course, is 10 2 \u003d 100.

Now let's represent this expression as a logarithmic one. We get log 10 100 = 2. When solving logarithms, all actions practically converge to finding the degree to which the base of the logarithm must be entered in order to obtain a given number.

To accurately determine the value of an unknown degree, you must 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 mindset and knowledge of the multiplication table. However, larger values ​​will require a power table. It can be used even by those who do not understand anything at all in 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 in the cells, the values ​​of the numbers are determined, which 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 real 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 equation. For example, 3 4 =81 can be written as the logarithm of 81 to base 3, which is four (log 3 81 = 4). For negative powers the rules are the same: 2 -5 \u003d 1/32 we write in the form of a logarithm, we get log 2 (1/32) \u003d -5. One of the most fascinating sections of mathematics is the topic of "logarithms". We will consider examples and solutions of equations a little lower, immediately after studying their properties. Now let's look at what inequalities look like and how to distinguish them from equations.

An expression of the following form is given: log 2 (x-1) > 3 - it is a logarithmic inequality, since the unknown value "x" is under the sign of the logarithm. And also in the expression two quantities are compared: the logarithm of the desired number in 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 of 2 x = √9) imply one or more specific numerical values ​​in the answer, while when solving the inequality, both the range of acceptable values ​​and the points breaking this function. As a consequence, the answer is not a simple set of individual numbers, as in the answer of the equation, but a continuous series or set of numbers.

Basic theorems about logarithms

When solving primitive tasks on 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 get acquainted with examples of equations later, let's first analyze each property in more detail.

1. The basic identity looks like this: a logaB =B. It only applies if a is greater than 0, not equal to one, and B is greater than zero.
2. The logarithm of the product can be represented in the following formula: log d (s 1 * s 2) = log d s 1 + log d s 2. In this case, the prerequisite is: d, s 1 and s 2 > 0; a≠1. You can give a proof for this formula of logarithms, with examples and a 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 get that s 1 *s 2 = a f1 *a f2 = a f1+f2 (degree properties ), and further by definition: log a (s 1 *s 2)= f 1 + f 2 = log a s1 + log a s 2, which was to be proved.
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 "property of the degree of the logarithm". It resembles the properties of ordinary degrees, and it is not surprising, because all mathematics rests on regular postulates. Let's look at the proof.

Let log a b \u003d t, it turns out a t \u003d b. If you raise both parts to the power m: a tn = b n ;

but since a tn = (a q) nt/q = b n , hence 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 logarithm problems are examples of equations and inequalities. They are found in almost all problem books, and are also included in the mandatory part of exams in mathematics. To enter a university or pass entrance tests in mathematics, you need to know how to solve such tasks correctly.

Unfortunately, a single plan or scheme to address and determine unknown value there is no logarithm, however, certain rules can be applied to each mathematical inequality or logarithmic equation. First of all, you should find out whether the expression can be simplified or reduced to general view. You can simplify long logarithmic expressions if you use their properties correctly. Let's get to know them soon.

When solving logarithmic equations, it is necessary to determine what kind of logarithm we have before us: an example of an expression may contain a natural logarithm or a decimal one.

Here are examples ln100, ln1026. Their solution boils down to the fact that you need to determine the degree to which the base 10 will be equal to 100 and 1026, respectively. For solutions of natural logarithms, one must 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 main theorems on logarithms.

1. The property of the logarithm of the product can be used in tasks where it is necessary to decompose a large value of the number 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 degree of the logarithm, we managed to solve at first glance a complex and unsolvable expression. It is only necessary to factorize the base and then take the exponent values ​​out of the sign of the logarithm.

Tasks from the exam

Logarithms are often found in entrance exams, especially a lot of logarithmic problems in the exam ( State exam for all high school graduates). Usually these tasks are present not only in part A (the easiest test part of the exam), but also in part C (the most difficult and voluminous tasks). The exam implies an accurate and perfect knowledge of the topic "Natural logarithms".

Examples and problem solutions are taken from official USE 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.

• All logarithms are best reduced to the same base so that the solution is not cumbersome and confusing.
• All expressions under the sign of the logarithm are indicated as positive, therefore, when taking out the exponent of the exponent of the expression, which is under the sign of the logarithm and as its base, the expression remaining under the logarithm must be positive.

The section of logarithms is of great importance in school course « Mathematical analysis". Tasks for logarithmic functions are based on other principles than tasks for inequalities and equations. Knowledge of the definitions and basic properties of the concepts of logarithm and logarithmic function, will ensure the successful solution of typical problems of the exam.

Before proceeding to explain what a logarithmic function is, it is worth referring to the definition of a logarithm.

Let's analyze specific example: and log a x = x, where a › 0, a ≠ 1.

The main properties of logarithms can be listed in several points:

Logarithm

Logarithm is a mathematical operation that allows using the properties of a concept to find the logarithm of a number or expression.

Examples:

Logarithm function and its properties

The logarithmic function has the form

We note right away that the graph of a function can be increasing for a › 1 and decreasing for 0 ‹ a ‹ 1. Depending on this, the function curve will have one form or another.

Here are the properties and method for plotting graphs of logarithms:

• the domain of f(x) is the set of all positive numbers, i.e. x can take any value from the interval (0; + ∞);
• ODZ functions - the set of all real numbers, i.e. y can be equal to any number from the interval (- ∞; +∞);
• if the base of the logarithm a > 1, then f(x) increases over the entire domain of definition;
• if the base of the logarithm is 0 ‹ a ‹ 1, then F is decreasing;
• the logarithmic function is neither even nor odd;
• the graph curve always passes through the point with coordinates (1;0).

Building both types of graphs is very simple, let's look at the process using an example

First you need to remember the properties simple logarithm and its functions. With their help, you need to build a table for specific x and y values. Then, on the coordinate axis, the obtained points should be marked and connected by a smooth line. This curve will be the required graph.

The logarithmic function is the inverse of exponential function, given by the formula y= a x . To verify this, it is enough to draw both curves on the same coordinate axis.

Obviously, both lines are mirror images of each other. By constructing a straight line y = x, you can see the axis of symmetry.

In order to quickly find the answer to the problem, you need to calculate the values ​​of the points for y = log 2⁡ x, and then simply move the origin of the coordinate points three divisions down the OY axis and 2 divisions to the left along the OX axis.

As proof, we will build a calculation table for the points of the graph y = log 2 ⁡ (x + 2) -3 and compare the obtained values ​​​​with the figure.

As you can see, the coordinates from the table and the points on the graph match, therefore, the transfer along the axes was carried out correctly.

Examples of solving typical USE problems

Most of the test tasks can be divided into two parts: finding the domain of definition, specifying the type of function according to the graph drawing, determining whether the function is increasing / decreasing.

For a quick answer to tasks, it is necessary to clearly understand that f (x) increases if the exponent of the logarithm a > 1, and decreases - when 0 ‹ a ‹ 1. However, not only the base, but also the argument can greatly affect the form of the function curve.

F(x) marked with a check mark are the correct answers. Doubts in this case are caused by examples 2 and 3. The “-” sign in front of log changes increasing to decreasing and vice versa.

Therefore, the graph y=-log 3⁡ x decreases over the entire domain of definition, and y= -log (1/3) ⁡x increases, despite the fact that the base is 0 ‹ a ‹ 1.

Answer: 3,4,5.

Answer: 4.

These types of tasks are considered easy and are estimated at 1-2 points.

Task 3.

Determine whether the function is decreasing or increasing and indicate the scope of its definition.

Y = log 0.7 ⁡(0.1x-5)

Since the base of the logarithm is less than one but greater than zero, the function of x is decreasing. According to the properties of the logarithm, the argument must also be greater than zero. Let's solve the inequality:

Answer: the domain of definition D(x) is the interval (50; + ∞).

Answer: 3, 1, OX axis, to the right.

Such tasks are classified as average and are estimated at 3-4 points.

Task 5. Find the range for a function:

It is known from the properties of the logarithm that the argument can only be positive. Therefore, we calculate the area of ​​​​admissible values ​​of the function. To do this, it will be necessary to solve a system of two inequalities.