Solving logarithmic equations. Methods for solving logarithmic equations

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Examples:

\(\log_(2)(⁡x) = 32\)
\(\log_3⁡x=\log_3⁡9\)
\(\log_3⁡((x^2-3))=\log_3⁡((2x))\)
\(\log_(x+1)((x^2+3x-7))=2\)
\(\lg^2⁡((x+1))+10=11 \lg⁡((x+1))\)

How to solve logarithmic equations:

When solving a logarithmic equation, you should strive to transform it to the form \(\log_a⁡(f(x))=\log_a⁡(g(x))\), and then make the transition to \(f(x)=g(x) \).

\(\log_a⁡(f(x))=\log_a⁡(g(x))\) \(⇒\) \(f(x)=g(x)\).

Example:\(\log_2⁡(x-2)=3\)

Very important! This transition can only be made if:

You have written for the original equation, and at the end you will check whether those found are included in the DL. If this is not done, extra roots may appear, which means a wrong decision.

The number (or expression) on the left and right is the same;

The logarithms on the left and right are “pure”, that is, there should be no multiplications, divisions, etc. – only single logarithms on either side of the equal sign.

For example:

Note that equations 3 and 4 can be easily solved by applying required properties logarithms.

Example . Solve the equation \(2\log_8⁡x=\log_8⁡2.5+\log_8⁡10\)

Solution :

Answer : \(5\)

Example : Solve the equation \(\log^2_2⁡(x)-3 \log_2(⁡x)+2=0\)

Solution :

Answer : \(4\); \(2\).

Logarithmic equations. From simple to complex.

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

What is a logarithmic equation?

This is an equation with logarithms. I’m surprised, right?) Then I’ll clarify. This is an equation in which the unknowns (x's) and expressions with them are found inside logarithms. And only there! This is important.

Here are some examples logarithmic equations :

log 3 x = log 3 9

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

log x+1 (x 2 +3x-7) = 2

lg 2 (x+1)+10 = 11lg(x+1)

Well, you understand... )

Pay attention! The most diverse expressions with X's are located exclusively within logarithms. If, suddenly, an X appears somewhere in the equation outside, For example:

log 2 x = 3+x,

this will be an equation mixed type. Such equations do not have clear rules for solving them. We will not consider them for now. By the way, there are equations where inside the logarithms only numbers. For example:

What can I say? You're lucky if you come across this! Logarithm with numbers is some number. That's all. It is enough to know the properties of logarithms to solve such an equation. Knowledge special rules, techniques adapted specifically to solve logarithmic equations, not required here.

So, what is a logarithmic equation- figured it out.

How to solve logarithmic equations?

Solution logarithmic equations- the thing is actually not very simple. So our section is a four... A decent amount of knowledge on all sorts of related topics is required. In addition, there is a special feature in these equations. And this feature is so important that it can safely be called the main problem in solving logarithmic equations. We will deal with this problem in detail in the next lesson.

For now, don't worry. We'll go the right way from simple to complex. On specific examples. The main thing is to delve into simple things and don’t be lazy to follow the links, I put them there for a reason... And everything will work out for you. Necessarily.

Let's start with the most elementary, simplest equations. To solve them, it is advisable to have an idea of ​​the logarithm, but nothing more. Just no idea logarithm, take on a decision logarithmic equations - somehow even awkward... Very bold, I would say).

The simplest logarithmic equations.

These are equations of the form:

1. log 3 x = log 3 9

2. log 7 (2x-3) = log 7 x

3. log 7 (50x-1) = 2

Solution process any logarithmic equation consists in the transition from an equation with logarithms to an equation without them. In the simplest equations this transition is carried out in one step. That's why they are the simplest.)

And such logarithmic equations are surprisingly easy to solve. See for yourself.

Let's solve the first example:

log 3 x = log 3 9

To solve this example, you don’t need to know almost anything, yes... Purely intuition!) What do we need especially don't like this example? What-what... I don't like logarithms! Right. So let's get rid of them. We look closely at the example, and a natural desire arises in us... Downright irresistible! Take and throw out logarithms altogether. And what’s good is that Can do! Mathematics allows. Logarithms disappear the answer is:

Great, right? This can (and should) always be done. Eliminating logarithms in this manner is one of the main ways to solve logarithmic equations and inequalities. In mathematics this operation is called potentiation. Of course, there are rules for such liquidation, but they are few. Remember:

You can eliminate logarithms without any fear if they have:

a) the same numerical bases

c) logarithms from left to right are pure (without any coefficients) and are in splendid isolation.

Let me clarify the last point. In the equation, let's say

log 3 x = 2log 3 (3x-1)

Logarithms cannot be removed. The two on the right doesn't allow it. The coefficient, you know... In the example

log 3 x+log 3 (x+1) = log 3 (3+x)

It is also impossible to potentiate the equation. There is no lone logarithm on the left side. There are two of them.

In short, you can remove logarithms if the equation looks like this and only like this:

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

In parentheses, where there is an ellipsis, there may be any expressions. Simple, super complex, all kinds. Whatever. The important thing is that after eliminating logarithms we are left with simpler equation. It is assumed, of course, that you already know how to solve linear, quadratic, fractional, exponential and other equations without logarithms.)

Now you can easily solve the second example:

log 7 (2x-3) = log 7 x

Actually, it’s decided in the mind. We potentiate, we get:

Well, is it very difficult?) As you can see, logarithmic part of the solution to the equation is only in eliminating logarithms... And then comes the solution to the remaining equation without them. A trivial matter.

Let's solve the third example:

log 7 (50x-1) = 2

We see that there is a logarithm on the left:

Let us remember that this logarithm is a number to which the base must be raised (i.e. seven) to obtain a sublogarithmic expression, i.e. (50x-1).

But this number is two! According to Eq. So:

That's basically all. Logarithm disappeared, What remains is a harmless equation:

We solved this logarithmic equation based only on the meaning of the logarithm. Is it still easier to eliminate logarithms?) I agree. By the way, if you make a logarithm from two, you can solve this example through elimination. Any number can be made into a logarithm. Moreover, the way we need it. Very useful trick in solving logarithmic equations and (especially!) inequalities.

Don't know how to make a logarithm from a number!? It's OK. Section 555 describes this technique in detail. You can master it and use it to the fullest! It greatly reduces the number of errors.

The fourth equation is solved in a completely similar way (by definition):

That's it.

Let's summarize this lesson. We looked at the solution of the simplest logarithmic equations using examples. This is very important. And not only because such equations appear in tests and exams. The fact is that even the most evil and complicated equations are necessarily reduced to the simplest!

Actually, the simplest equations are the final part of the solution any equations. And this final part must be understood strictly! And one more thing. Be sure to read this page to the end. There's a surprise there...)

Now we decide for ourselves. Let's get better, so to speak...)

Find the root (or sum of roots, if there are several) of the equations:

ln(7x+2) = ln(5x+20)

log 2 (x 2 +32) = log 2 (12x)

log 16 (0.5x-1.5) = 0.25

log 0.2 (3x-1) = -3

ln(e 2 +2x-3) = 2

log 2 (14x) = log 2 7 + 2

Answers (in disarray, of course): 42; 12; 9; 25; 7; 1.5; 2; 16.

What, not everything works out? Happens. Don't worry! Section 555 explains the solution to all of these examples in a clear and detailed manner. You'll definitely figure it out there. You will also learn useful practical techniques.

Everything worked out!? All examples of “one left”?) Congratulations!

It's time to reveal the bitter truth to you. Successful solving of these examples does not guarantee success in solving all other logarithmic equations. Even the simplest ones like these. Alas.

The fact is that the solution to any logarithmic equation (even the most elementary!) consists of two equal parts. Solving the equation and working with ODZ. We have mastered one part - solving the equation itself. It's not that hard right?

For this lesson, I specially selected examples in which DL does not affect the answer in any way. But not everyone is as kind as me, right?...)

Therefore, it is imperative to master the other part. ODZ. This is the main problem in solving logarithmic equations. And not because it’s difficult - this part is even easier than the first. But because they simply forget about ODZ. Or they don't know. Or both). And they fall out of the blue...

In the next lesson we will deal with this problem. Then you can confidently decide any simple logarithmic equations and approach quite solid tasks.

If you like this site...

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

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

We are all familiar with equations primary classes. There we also learned to solve the simplest examples, and we must admit that they find their application even in higher mathematics. Everything is simple with equations, including quadratic equations. If you are having trouble with this topic, we highly recommend that you review it.

You've probably already gone through logarithms too. However, we consider it important to tell what it is for those who do not yet know. A logarithm is equated to the power to which the base must be raised to obtain the number to the right of the logarithm sign. Let's give an example based on which everything will become clear to you.

If you raise 3 to the fourth power, you get 81. Now substitute the numbers by analogy, and you will finally understand how logarithms are solved. Now all that remains is to combine the two concepts discussed. Initially, the situation seems extremely complicated, but upon closer examination the weight falls into place. We are sure that after this short article you will not have problems in this part of the Unified State Exam.

Today there are many ways to solve such structures. We will tell you about the simplest, most effective and most applicable in the case of Unified State Examination tasks. Solving logarithmic equations must start from the very beginning. simple example. The simplest logarithmic equations consist of a function and one variable in it.

It's important to note that x is inside the argument. A and b must be numbers. In this case, you can simply express the function in terms of a number to a power. It looks like this.

Of course, solving a logarithmic equation using this method will lead you to the correct answer. The problem for the vast majority of students in this case is that they do not understand what comes from and where it comes from. As a result, you have to put up with mistakes and not get the desired points. The most offensive mistake will be if you mix up the letters. To solve an equation this way, you need to memorize this standard school formula because it is difficult to understand.

To make it easier, you can resort to another method - the canonical form. The idea is extremely simple. Turn your attention back to the problem. Remember that the letter a is a number, not a function or variable. A is not equal to one and greater than zero. There are no restrictions on b. Now, of all the formulas, let us remember one. B can be expressed as follows.

It follows from this that everything original equations with logarithms can be represented as:

Now we can drop the logarithms. It will work out simple design, which we have already seen earlier.

The convenience of this formula lies in the fact that it can be used in the most different cases, and not just for the simplest designs.

Don't worry about OOF!

Many experienced mathematicians will notice that we have not paid attention to the domain of definition. The rule boils down to the fact that F(x) is necessarily greater than 0. No, we did not miss this point. Now we are talking about another serious advantage of the canonical form.

There will be no extra roots here. If a variable will only appear in one place, then a scope is not necessary. It is done automatically. To verify this judgment, try solving several simple examples.

How to solve logarithmic equations with different bases

These are already complex logarithmic equations, and the approach to solving them must be special. Here it is rarely possible to limit ourselves to the notorious canonical form. Let's start our detailed story. We have the following construction.

Pay attention to the fraction. It contains the logarithm. If you see this in a task, it’s worth remembering one interesting trick.

What does it mean? Each logarithm can be represented as the quotient of two logarithms with a convenient base. And this formula has special case, which is applicable with this example (meaning if c=b).

This is exactly the fraction we see in our example. Thus.

Essentially, we turned the fraction around and got a more convenient expression. Remember this algorithm!

Now we need that the logarithmic equation did not contain different reasons. Let's represent the base as a fraction.

In mathematics there is a rule based on which you can derive a degree from a base. The following construction results.

It would seem that what is stopping us from now turning our expression into canonical form and just solve it? It's not that simple. There should be no fractions before the logarithm. Let's fix this situation! A fraction is allowed to be used as a degree.

Respectively.

If the bases are the same, we can remove the logarithms and equate the expressions themselves. This way the situation will become much simpler than it was. Will remain elementary equation, which each of us knew how to solve back in 8th or even 7th grade. You can do the calculations yourself.

We have obtained the only true root of this logarithmic equation. Examples of solving a logarithmic equation are quite simple, aren't they? Now you will be able to deal with even the most difficult problems on your own. complex tasks for preparing and passing the Unified State Exam.

What's the result?

In the case of any logarithmic equations, we start from one very important rule. It is necessary to act in such a way as to bring the expression to the maximum simple view. In this case you will have more chances not only solve the task correctly, but also do it in the simplest and most logical way possible. This is exactly how mathematicians always work.

We strongly do not recommend that you look for difficult paths, especially in this case. Remember a few simple rules, which will allow you to transform any expression. For example, reduce two or three logarithms to the same base or derive a power from the base and win on this.

It is also worth remembering that solving logarithmic equations requires constant practice. Gradually you will move on to more and more complex structures, and this will lead you to confidently solving all variants of problems on the Unified State Exam. Prepare well in advance for your exams, and good luck!

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. under the title “Description of an amazing table of logarithms”, Napier’s theory of logarithms was given in a fairly complete volume, the method of calculating logarithms was given as the simplest, therefore Napier’s merits in the invention of logarithms were 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 these “ artificial numbers“to call in one word “logarithm”, which in translation from Greek is “correlated numbers”, taken one from an arithmetic progression, and the other from a geometric progression specially selected for it. The first tables in Russian were published in 1703. with the participation of a wonderful teacher of the 18th century. L. F. Magnitsky. In the development of the theory of logarithms great value had the works of St. Petersburg academician Leonhard Euler. 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 ancients mathematical problems Mesopotamia, India, China, Greece, unknown quantities expressed the number of peacocks in the garden, the number of bulls in the herd, the totality of things taken into account when dividing property. Scribes, officials and initiates well trained in the science of accounts secret knowledge The priests coped with such tasks quite successfully.

Sources that have reached us indicate that ancient scientists owned some general techniques solving problems with unknown quantities. However, not a single papyrus, not a single clay tablet no description of these techniques is given. The authors only occasionally provided 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. Logarithm of the quotient of two positive numbers equal to the difference logarithms of dividend and divisor

Comment. If

P4. Logarithm of degree positive number equal to the product exponent per 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 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.