Logarithms: examples and solutions. Learning to solve simple logarithmic equations

Solving logarithmic equations. Part 1.

Logarithmic equation is an equation in which the unknown is contained under the sign of the logarithm (in particular, in the base of the logarithm).

The simplest logarithmic equation has the form:

Solving any logarithmic equation involves a transition from logarithms to expressions under the sign of logarithms. However, this action expands the scope acceptable values equation and can lead to the appearance of extraneous roots. To avoid the appearance of foreign roots, you can do one of three ways:

1. Make an equivalent transition from the original equation to a system including

depending on which inequality or simpler.

If the equation contains an unknown in the base of the logarithm:

then we go to the system:

2. Separately find the range of acceptable values ​​of the equation, then solve the equation and check whether the solutions found satisfy the equation.

3. Solve the equation, and then check: substitute the found solutions into the original equation and check whether we get the correct equality.

Logarithmic equation of any level of complexity ultimately always comes down to a simple logarithmic equation.

All logarithmic equations can be divided into four types:

1 . Equations that contain logarithms only to the first power. With the help of transformations and use, they are brought to the form

Example. Let's solve the equation:

Let us equate the expressions under the logarithm sign:

Let's check whether our root of the equation satisfies:

Yes, it satisfies.

Answer: x=5

2 . Equations that contain logarithms to powers other than 1 (particularly in the denominator of a fraction). Such equations can be solved using introducing a change of variable.

Example. Let's solve the equation:

Let's find the ODZ equation:

The equation contains logarithms squared, so it can be solved using a change of variable.

Important! Before introducing a replacement, you need to “pull apart” the logarithms that are part of the equation into “bricks”, using the properties of logarithms.

When “pulling apart” logarithms, it is important to use the properties of logarithms very carefully:

In addition, there is one more subtle point here, and in order to avoid a common mistake, we will use an intermediate equality: we will write the degree of the logarithm in this form:

Likewise,

Let's substitute the resulting expressions into the original equation. We get:

Now we see that the unknown is contained in the equation as part of . Let's introduce the replacement: . Since it can take any real value, we do not impose any restrictions on the variable.

Preparation for the final test in mathematics includes an important section - “Logarithms”. Tasks from this topic are necessarily contained in the Unified State Examination. Experience from past years shows that logarithmic equations have caused difficulties for many schoolchildren. Therefore, students with different levels of training must understand how to find the correct answer and quickly cope with them.

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When preparing for the Unified State Exam, high school graduates need a reliable source that provides the most complete and accurate information for successfully solving test problems. However, the textbook is not always at hand, and searching necessary rules and formulas on the Internet often takes time.

The Shkolkovo educational portal allows you to prepare for the Unified State Exam anywhere at any time. Our website offers the most convenient approach to repeating and assimilating a large amount of information on logarithms, as well as with one and several unknowns. Start with easy equations. If you cope with them without difficulty, move on to more complex ones. If you have trouble solving a particular inequality, you can add it to your Favorites so you can return to it later.

You can find the necessary formulas to complete the task, repeat special cases and methods for calculating the root of a standard logarithmic equation by looking at the “Theoretical Help” section. Shkolkovo teachers collected, systematized and outlined everything necessary for successful completion materials in the simplest and most understandable form.

In order to easily cope with tasks of any complexity, on our portal you can familiarize yourself with the solution of some standard logarithmic equations. To do this, go to the “Catalogues” section. We present large number examples, including profile equations Unified State Exam level in mathematics.

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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 negative number(that is, any positive) “b” by its base “a” is considered to be the power of “c” to which the base “a” must be raised in order to ultimately obtain 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:

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 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 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 answers. numerical values, while when solving the inequality, both the range of permissible values ​​and the breakpoints of this function are determined. 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, 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.
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. 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 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 natural logarithms need to apply logarithmic identities or their properties. Let's look at the solution with examples logarithmic problems different 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). Typically, these tasks are present not only in part A (the easiest test part of the 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 the official versions of the Unified State Exam. 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.

Logarithmic equation is an equation in which the unknown (x) and expressions with it are under the sign of the logarithmic function. Solving logarithmic equations assumes that you are already familiar with and .
How to solve logarithmic equations?

The simplest equation is log a x = b, where a and b are some numbers, x is an unknown.
Solving a logarithmic equation is x = a b provided: a > 0, a 1.

It should be noted that if x is somewhere outside the logarithm, for example log 2 x = x-2, then such an equation is already called mixed and a special approach is needed to solve it.

The ideal case is when you come across an equation in which only numbers are under the logarithm sign, for example x+2 = log 2 2. Here it is enough to know the properties of logarithms to solve it. But such luck does not happen often, so get ready for more difficult things.

But first, let's start with simple equations. To solve them, it is desirable to have the most general idea about the logarithm.

Solving simple logarithmic equations

These include equations of the type log 2 x = log 2 16. The naked eye can see that by omitting the sign of the logarithm we get x = 16.

To solve a more complex logarithmic equation, it is usually reduced to solving the usual algebraic equation or to the solution of the simplest logarithmic equation log a x = b. In the simplest equations this happens in one movement, which is why they are called simplest.

The above method of dropping logarithms is one of the main ways to solve logarithmic equations and inequalities. In mathematics, this operation is called potentiation. There are certain rules or restrictions for this type of operation:

• logarithms have the same numerical bases
• The logarithms in both sides of the equation are free, i.e. without any coefficients or other various kinds of expressions.

Let's say in the equation log 2 x = 2log 2 (1 - x) potentiation is not applicable - the coefficient 2 on the right does not allow it. In the following example, log 2 x+log 2 (1 - x) = log 2 (1+x) also does not satisfy one of the restrictions - there are two logarithms on the left. If there was only one, it would be a completely different matter!

In general, you can remove logarithms only if the equation has the form:

log a (...) = log a (...)

Absolutely any expressions can be placed in brackets; this has absolutely no effect on the potentiation operation. And after eliminating the logarithms, a simpler equation will remain - linear, quadratic, exponential, etc., which, I hope, you already know how to solve.

Let's take another example:

log 3 (2x-5) = log 3 x

We apply potentiation, we get:

log 3 (2x-1) = 2

Based on the definition of a logarithm, namely, that a logarithm is the number to which the base must be raised in order to obtain an expression that is under the logarithm sign, i.e. (4x-1), we get:

Again we received a beautiful answer. Here we did without eliminating logarithms, but potentiation is also applicable here, because a logarithm can be made from any number, and exactly the one we need. This method is very helpful in solving logarithmic equations and especially inequalities.

Let's solve our logarithmic equation log 3 (2x-1) = 2 using potentiation:

Let's imagine the number 2 as a logarithm, for example, this log 3 9, because 3 2 =9.

Then log 3 (2x-1) = log 3 9 and again we get the same equation 2x-1 = 9. I hope everything is clear.

So we looked at how to solve the simplest logarithmic equations, which are actually very important, because solving logarithmic equations, even the most terrible and twisted ones, in the end always comes down to solving the simplest equations.

In everything we did above, we missed one very important point, which will play a decisive role in the future. The fact is that the solution to any logarithmic equation, even the most elementary one, consists of two equal parts. The first is the solution of the equation itself, the second is working with the range of permissible values ​​(APV). This is exactly the first part that we have mastered. In the above examples, ODZ does not affect the answer in any way, so we did not consider it.

Let's take another example:

log 3 (x 2 -3) = log 3 (2x)

Outwardly, this equation is no different from an elementary one, which can be solved very successfully. But this is not entirely true. No, of course we will solve it, but most likely incorrectly, because it contains a small ambush, into which both C-grade students and excellent students immediately fall into it. Let's take a closer look.

Let's say you need to find the root of the equation or the sum of the roots, if there are several of them:

log 3 (x 2 -3) = log 3 (2x)

We use potentiation, it is acceptable here. As a result, we obtain an ordinary quadratic equation.

Finding the roots of the equation:

It turned out two roots.

Answer: 3 and -1

At first glance everything is correct. But let's check the result and substitute it into the original equation.

Let's start with x 1 = 3:

log 3 6 = log 3 6

The check was successful, now the queue is x 2 = -1:

log 3 (-2) = log 3 (-2)

Okay, stop! On the outside everything is perfect. One thing - there are no logarithms from negative numbers! This means that the root x = -1 is not suitable for solving our equation. And therefore the correct answer will be 3, not 2, as we wrote.

This is where ODZ played its fatal role, which we had forgotten about.

Let me remind you that the range of acceptable values ​​includes those values ​​of x that are allowed or make sense for the original example.

Without ODZ, any solution, even an absolutely correct one, of any equation turns into a lottery - 50/50.

How could we get caught solving a seemingly elementary example? But precisely at the moment of potentiation. Logarithms disappeared, and with them all restrictions.

What to do in this case? Refuse to eliminate logarithms? And completely refuse to solve this equation?

No, we just, like real heroes from one famous song, will take a detour!

Before we begin solving any logarithmic equation, we will write down the ODZ. But after that, you can do whatever your heart desires with our equation. Having received the answer, we simply throw out those roots that are not included in our ODZ and write down the final version.

Now let’s decide how to record ODZ. To do this, we carefully examine the original equation and look for suspicious places in it, such as division by x, even root, etc. Until we have solved the equation, we do not know what x is equal to, but we know for sure that those x that, when substituted, give division by 0 or the square root of a negative number, are obviously not suitable as an answer. Therefore, such x are unacceptable, while the rest will constitute ODZ.

Let's use the same equation again:

log 3 (x 2 -3) = log 3 (2x)

log 3 (x 2 -3) = log 3 (2x)

As you can see, there is no division by 0, square roots also not, but there are expressions with x in the body of the logarithm. Let us immediately remember that the expression inside the logarithm must always be >0. We write this condition in the form of ODZ:

Those. We haven’t solved anything yet, but we have already written down a mandatory condition for the entire sublogarithmic expression. The curly brace means that these conditions must be true simultaneously.

The ODZ is written down, but it is also necessary to solve the resulting system of inequalities, which is what we will do. We get the answer x > v3. Now we know for sure which x will not suit us. And then we begin to solve the logarithmic equation itself, which is what we did above.

Having received the answers x 1 = 3 and x 2 = -1, it is easy to see that only x1 = 3 suits us, and we write it down as the final answer.

For the future, it is very important to remember the following: we solve any logarithmic equation in 2 stages. The first is to solve the equation itself, the second is to solve the ODZ condition. Both stages are performed independently of each other and are compared only when writing the answer, i.e. discard everything unnecessary and write down the correct answer.

To reinforce the material, we strongly recommend watching the video:

The video shows other examples of solving log. equations and working out the interval method in practice.

To this question, how to solve logarithmic equations That's all for now. If something is decided by the log. equations remain unclear or incomprehensible, write your questions in the comments.

Note: The Academy of Social Education (ASE) is ready to accept new students.

Introduction

Logarithms were invented to speed up and simplify calculations. The idea of ​​a logarithm, that is, the idea of ​​expressing numbers as powers of the same base, belongs to Mikhail Stiefel. But in Stiefel’s time, mathematics was not so developed and the idea of ​​the logarithm was not developed. Logarithms were later invented simultaneously and independently of each other by the Scottish scientist John Napier (1550-1617) and the Swiss Jobst Burgi (1552-1632). Napier was the first to publish the work in 1614. entitled "Description of the amazing table of logarithms", Napier's theory of logarithms was given in sufficient detail in full, the method for calculating logarithms is given the simplest, therefore Napier’s merits in the invention of logarithms are greater than those of Bürgi. Bürgi worked on the tables at the same time as Napier, but for a long time kept them secret and published them only in 1620. Napier mastered the idea of ​​the logarithm around 1594. although the tables were published 20 years later. At first he called his logarithms “artificial numbers” and only then proposed to call these “artificial numbers” in one word “logarithm”, which translated from Greek means “correlated numbers”, taken one from an arithmetic progression, and the other from a geometric progression specially selected for it. progress. The first tables in Russian were published in 1703. with the participation of a wonderful teacher of the 18th century. L. F. Magnitsky. The works of St. Petersburg academician Leonhard Euler were of great importance in the development of the theory of logarithms. He was the first to consider logarithms as the inverse of raising to a power; he introduced the terms “logarithm base” and “mantissa.” Briggs compiled tables of logarithms with base 10. Decimal tables are more convenient for practical use, their theory is simpler than that of Napier’s logarithms . That's why decimal logarithms sometimes called brigs. The term "characterization" was introduced by Briggs.

In those distant times, when the sages first began to think about equalities containing unknown quantities, there were probably no coins or wallets. But there were heaps, as well as pots and baskets, which were perfect for the role of storage caches that could hold an unknown number of items. In the ancient mathematical problems of Mesopotamia, India, China, Greece, unknown quantities expressed the number of peacocks in the garden, the number of bulls in the herd, and the totality of things taken into account when dividing property. Scribes, officials and priests initiated into secret knowledge, well trained in the science of accounts, coped with such tasks quite successfully.

Sources that have reached us indicate that ancient scientists had some general techniques for solving problems with unknown quantities. However, not a single papyrus or clay tablet contains a description of these techniques. The authors only occasionally supplied their numerical calculations with skimpy comments such as: “Look!”, “Do this!”, “You found the right one.” In this sense, the exception is the “Arithmetic” of the Greek mathematician Diophantus of Alexandria (III century) - a collection of problems for composing equations with a systematic presentation of their solutions.

However, the first manual for solving problems that became widely known was the work of the Baghdad scientist of the 9th century. Muhammad bin Musa al-Khwarizmi. The word "al-jabr" from the Arabic name of this treatise - "Kitab al-jaber wal-mukabala" ("Book of restoration and opposition") - over time turned into the well-known word "algebra", and the work of al-Khwarizmi itself served the starting point in the development of the science of solving equations.

Logarithmic equations and inequalities

1. Logarithmic equations

An equation containing an unknown under the logarithm sign or at its base is called a logarithmic equation.

The simplest logarithmic equation is an equation of the form

log a x = b . (1)

Statement 1. If a > 0, a≠ 1, equation (1) for any real b has the only solution x = a b .

Example 1. Solve the equations:

a)log 2 x= 3, b) log 3 x= -1, c)

Solution. Using Statement 1, we obtain a) x= 2 3 or x= 8; b) x= 3 -1 or x= 1 / 3 ; c)

Let us present the basic properties of the logarithm.

P1. Basic logarithmic identity:

Where a > 0, a≠ 1 and b > 0.

P2. Logarithm of the product of positive factors equal to the sum logarithms of these factors:

log a N 1 · N 2 = log a N 1 + log a N 2 (a > 0, a ≠ 1, N 1 > 0, N 2 > 0).

Comment. If N 1 · N 2 > 0, then property P2 takes the form

log a N 1 · N 2 = log a |N 1 | + log a |N 2 | (a > 0, a ≠ 1, N 1 · N 2 > 0).

P3. The logarithm of the quotient of two positive numbers is equal to the difference between the logarithms of the dividend and the divisor

Comment. If

P4. The logarithm of the power of a positive number is equal to the product of the exponent and the logarithm of this number:

log a N k = k log a N (a > 0, a ≠ 1, N > 0).

Comment. If k- even number ( k = 2s), That

log a N 2s = 2s log a |N | (a > 0, a ≠ 1, N ≠ 0).

P5. Formula for moving to another base:

in particular if N = b, we get

Using properties P4 and P5, it is easy to obtain following properties

and, if in (5) c- even number ( c = 2n), holds

Let us list the main properties of the logarithmic function f (x) = log a x :

1. The domain of definition of a logarithmic function is the set of positive numbers.

2. The range of values ​​of the logarithmic function is the set of real numbers.

3. When a> 1 logarithmic function is strictly increasing (0< x 1 < x 2log a x 1 < loga x 2), and at 0< a < 1, - строго убывает (0 < x 1 < x 2log a x 1 > log a x 2).

4. log a 1 = 0 and log a a = 1 (a > 0, a ≠ 1).

5. If a> 1, then the logarithmic function is negative when x(0;1) and positive at x(1;+∞), and if 0< a < 1, то логарифмическая функция положительна при x (0;1) and negative at x (1;+∞).

6. If a> 1, then the logarithmic function is convex upward, and if a(0;1) - convex downwards.

The following statements (see, for example,) are used when solving logarithmic equations.