exponential equations. As you know, the USE includes simple equations. We have already considered some - these are logarithmic, trigonometric, rational. Here are exponential equations.

In a recent article, we worked with exponential expressions, it will be useful. The equations themselves are solved simply and quickly. It is only required to know the properties of the exponents and ... About thisFurther.

We list the properties of exponents:

The zero power of any number is equal to one.

Consequence of this property:

A little more theory.

An exponential equation is an equation containing a variable in the indicator, that is, this equation is of the form:

f(x) an expression that contains a variable

Methods for solving exponential equations

1. As a result of transformations, the equation can be reduced to the form:

Then we apply the property:

2. When obtaining an equation of the form a f (x) = b the definition of the logarithm is used, we get:

3. As a result of the transformations, you can get an equation of the form:

The logarithm is applied:

Express and find x.

In tasks USE options it will be enough to use the first method.

That is, it is necessary to present the left and right parts as degrees with the same base, and then we equate the indicators and solve the usual linear equation.

Consider the equations:

Find the root of Equation 4 1-2x = 64.

It is necessary to make sure that in the left and right parts were exponential expressions with one base. We can represent 64 as 4 to the power of 3. We get:

4 1–2x = 4 3

1 - 2x = 3

– 2x = 2

x = - 1

Examination:

4 1–2 (–1) = 64

4 1 + 2 = 64

4 3 = 64

64 = 64

Answer: -1

Find the root of equation 3 x-18 = 1/9.

It is known that

So 3 x-18 = 3 -2

The bases are equal, we can equate the indicators:

x - 18 \u003d - 2

x = 16

Examination:

3 16–18 = 1/9

3 –2 = 1/9

1/9 = 1/9

Answer: 16

Find the root of the equation:

Let's represent the fraction 1/64 as one fourth to the third power:

2x - 19 = 3

2x = 22

x = 11

Examination:

Answer: 11

Find the root of the equation:

Let's represent 1/3 as 3 -1, and 9 as 3 squared, we get:

(3 –1) 8–2x = 3 2

3 –1∙(8–2х) = 3 2

3 -8 + 2x \u003d 3 2

Now we can equate the indicators:

– 8+2x = 2

2x = 10

x = 5

Examination:

Answer: 5

26654. Find the root of the equation:

Decision:

Answer: 8.75

Indeed, to whatever degree we raise positive number a, there is no way we can get a negative number.

Any exponential equation after appropriate transformations is reduced to solving one or more simple ones.In this section, we will also consider the solution of some equations, do not miss it!That's all. Good luck to you!

Sincerely, Alexander Krutitskikh.

P.S: I would be grateful if you tell about the site in social networks.

Equipment:

• a computer,
• multimedia projector,
• screen,
• Appendix 1(slide presentation in PowerPoint) “Methods for solving exponential equations”
• Appendix 2(The solution of the equation of the type “Three different bases degrees" in Word)
• Annex 3(handout in Word for practical work).
• Appendix 4(handout in Word for homework).

During the classes

1. Organizational stage

• message of the topic of the lesson (written on the board),
• the need for a generalizing lesson in grades 10-11:

The stage of preparing students for the active assimilation of knowledge

Repetition

Definition.

An exponential equation is an equation containing a variable in the exponent (the student answers).

Teacher's note. The exponential equations belong to the class of transcendental equations. This hard-to-pronounce name suggests that such equations, generally speaking, cannot be solved in the form of formulas.

They can only be solved by approximately numerical methods on computers. But what about exam questions? The whole trick is that the examiner composes the problem in such a way that it just admits an analytical solution. In other words, you can (and should!) do such identical transformations that reduce the given exponential equation to the simplest exponential equation. This is the simplest equation and is called: the simplest exponential equation. It is solved logarithm.

The situation with the solution of an exponential equation resembles a journey through a maze, which was specially invented by the compiler of the problem. From these very general considerations, quite specific recommendations follow.

For successful solution exponential equations it is necessary:

1. Not only actively know all exponential identities, but also find sets of values ​​of the variable on which these identities are defined, so that when using these identities, one does not acquire unnecessary roots, and even more so, does not lose solutions to the equation.

2. Actively know all exponential identities.

3. Clearly, in detail and without errors, perform mathematical transformations of equations (transfer terms from one part of the equation to another, not forgetting to change the sign, reduce the fraction to a common denominator, etc.). This is called mathematical culture. At the same time, the calculations themselves should be done automatically by hands, and the head should think about the general guiding thread of the solution. It is necessary to make transformations as carefully and in detail as possible. Only this will guarantee a correct, error-free solution. And remember: small arithmetic error can simply create a transcendental equation, which in principle cannot be solved analytically. It turns out that you lost your way and ran into the wall of the labyrinth.

4. Know the methods of solving problems (that is, know all the paths through the labyrinth of the solution). For correct orientation at each stage, you will have to (consciously or intuitively!):

• define equation type;
• remember the corresponding type solution method tasks.

The stage of generalization and systematization of the studied material.

The teacher, together with the students, with the involvement of a computer, conducts an overview repetition of all types of exponential equations and methods for solving them, draws up general scheme. (Using a tutorial computer program L.Ya. Borevsky "Course of Mathematics - 2000", the author of the presentation in PowerPoint - T.N. Kuptsov.)

Rice. one. The figure shows a general scheme of all types of exponential equations.

As can be seen from this diagram, the strategy for solving exponential equations is to reduce this exponential equation to the equation, first of all, with the same bases , and then - and with the same exponents.

Having obtained an equation with the same bases and exponents, you replace this degree with a new variable and get a simple algebraic equation (usually, fractional rational or quadratic) with respect to this new variable.

By solving this equation and making an inverse substitution, you end up with a set of simple exponential equations that are solved in general view using logarithms.

Equations stand apart in which only products of (private) powers occur. Using exponential identities, it is possible to bring these equations immediately to one base, in particular, to the simplest exponential equation.

Consider how an exponential equation with three different bases of degrees is solved.

(If the teacher has a teaching computer program by L.Ya. Borevsky "Course of Mathematics - 2000", then naturally we work with the disk, if not, you can print out this type of equation for each desk from it, presented below.)

Rice. 2. Equation solution plan.

Rice. 3. Beginning to solve the equation

Rice. 4. The end of the solution of the equation.

Doing practical work

Determine the type of equation and solve it.

1.
2.
3.	0,125
4.
5.
6.

Summing up the lesson

Grading a lesson.

end of lesson

For the teacher

Scheme of practical work answers.

Exercise: select equations from the list of equations specified type(Enter the answer number in the table):

1. Three different bases
2. Two different bases - different exponents
3. Bases of powers - powers of one number
4. Same bases, different exponents
5. Same exponent bases - same exponents
6. Product of powers
7. Two different bases of degrees - the same indicators
8. The simplest exponential equations

1. (product of powers)

2. (same bases - different exponents)

At the stage of preparation for the final testing, high school students need to improve their knowledge on the topic "Exponential Equations". The experience of past years indicates that such tasks cause certain difficulties for schoolchildren. Therefore, high school students, regardless of their level of preparation, need to carefully master the theory, memorize the formulas and understand the principle of solving such equations. Having learned to cope with this type of tasks, graduates will be able to count on high scores when passing the exam in mathematics.

Get ready for the exam testing together with Shkolkovo!

When repeating the materials covered, many students are faced with the problem of finding the formulas needed to solve the equations. The school textbook is not always at hand, and the selection necessary information on the topic on the Internet takes a long time.

The Shkolkovo educational portal invites students to use our knowledge base. We implement completely new method preparation for final testing. Studying on our site, you will be able to identify gaps in knowledge and pay attention to precisely those tasks that cause the greatest difficulties.

The teachers of "Shkolkovo" collected, systematized and presented everything necessary for a successful passing the exam material in the most simple and accessible form.

The main definitions and formulas are presented in the "Theoretical Reference" section.

For a better assimilation of the material, we recommend that you practice the assignments. Carefully review the examples of exponential equations with solutions presented on this page in order to understand the calculation algorithm. After that, proceed with the tasks in the "Catalogs" section. You can start with the easiest tasks or go straight to solving complex exponential equations with several unknowns or . The database of exercises on our website is constantly supplemented and updated.

Those examples with indicators that caused you difficulties can be added to the "Favorites". So you can quickly find them and discuss the solution with the teacher.

To successfully pass the exam, study on the Shkolkovo portal every day!

In this article, you will get acquainted with all types exponential equations and algorithms for solving them, learn to recognize what type exponential equation, which you need to solve, and apply the appropriate method to solve it. Detailed solution of examples exponential equations each type you can see in the corresponding VIDEO TUTORIALS.

An exponential equation is an equation in which the unknown is contained in the exponent.

Before you start solving the exponential equation, it is useful to do a few preliminary action , which can greatly facilitate the course of its solution. These are the actions:

1. Factorize all bases of powers into prime factors.

2. Present the roots as a degree.

3. Decimal fractions represent in the form of ordinary.

4. mixed numbers write as improper fractions.

You will realize the benefits of these actions in the process of solving equations.

Consider the main types exponential equations and algorithms for their solution.

1. Type equation

This equation is equivalent to the equation

Watch this VIDEO to solve the equation of this type.

2. Type equation

In equations of this type:

b) the coefficients for the unknown in the exponent are equal.

To solve this equation, you need to bracket the multiplier to the smallest degree.

An example of solving an equation of this type:

look at the VIDEO.

3. Type equation

These types of equations differ in that

a) all degrees have the same base

b) the coefficients for the unknown in the exponent are different.

Equations of this type are solved using a change of variables. Before introducing a replacement, it is desirable to get rid of free terms in the exponent. (, , etc)

Look in the VIDEO for the solution of this type of equation:

4. Homogeneous equations kind

Distinctive features of homogeneous equations:

a) all monomials have the same degree,

b) the free term is equal to zero,

c) the equation contains powers with two different bases.

Homogeneous equations are solved by a similar algorithm.

To solve this type of equation, divide both sides of the equation by (can be divided by or by )

Attention! When dividing the right and left sides of the equation by an expression containing an unknown, you can lose the roots. Therefore, it is necessary to check whether the roots of the expression by which we divide both parts of the equation are the roots of the original equation.

In our case, since the expression is not equal to zero for any values ​​of the unknown, we can divide by it without fear. We divide the left side of the equation by this expression term by term. We get:

Reduce the numerator and denominator of the second and third fractions:

Let's introduce a replacement:

And title="(!LANG:t>0">при всех !} allowed values unknown.

Get quadratic equation:

Let's solve the quadratic equation, find the values, which satisfy the condition title="(!LANG:t>0">, а затем вернемся к исходному неизвестному.!}

Watch in the VIDEO detailed solution homogeneous equation:

5. Type equation

When solving this equation, we will proceed from the fact that title="(!LANG:f(x)>0">!}

The original equality holds in two cases:

1. If , since 1 is equal to 1 to any power,

2. Under two conditions:

Title="(!LANG:delim(lbrace)(matrix(2)(1)((f(x)>0) (g(x)=h(x)) (x-8y+9z=0))) ( )">!}

Watch the VIDEO for a detailed solution of the equation

What is an exponential equation? Examples.

So, an exponential equation ... A new unique exhibit at our general exhibition of a wide variety of equations!) As it almost always happens, the keyword of any new mathematical term is the corresponding adjective that characterizes it. So here too. keyword in the term "exponential equation" is the word "demonstrative". What does it mean? This word means that the unknown (x) is in terms of any degree. And only there! This is extremely important.

For example, these simple equations:

3 x +1 = 81

5x + 5x +2 = 130

4 2 2 x -17 2 x +4 = 0

Or even these monsters:

2 sin x = 0.5

Please pay attention to one important thing: in grounds degrees (bottom) - only numbers. But in indicators degrees (top) - a wide variety of expressions with x. Absolutely any.) Everything depends on the specific equation. If, suddenly, x comes out in the equation somewhere else, in addition to the indicator (say, 3 x \u003d 18 + x 2), then such an equation will already be an equation mixed type . Such equations do not have clear rules for solving. Therefore, in this lesson we will not consider them. To the delight of the students.) Here we will consider only exponential equations in a "pure" form.

Generally speaking, even pure exponential equations are not clearly solved in all cases and not always. But among the rich variety of exponential equations, there are certain types which can and should be addressed. It is these types of equations that we will consider with you. And we will definitely solve the examples.) So we settle in comfortably and - on the road! As in computer "shooters", our journey will pass through the levels.) From elementary to simple, from simple to medium and from medium to complex. Along the way, you will also be waiting for a secret level - tricks and methods for solving non-standard examples. The ones you won't read about most school textbooks… Well, in the end, of course, waiting for you final boss like a home.)

Level 0. What is the simplest exponential equation? Solution of the simplest exponential equations.

To begin with, let's look at some frank elementary. You have to start somewhere, right? For example, this equation:

2 x = 2 2

Even without any theories, by simple logic and common sense it is clear that x = 2. There is no other way, right? No other value of x is good ... Now let's turn our attention to decision record this cool exponential equation:

2 x = 2 2

X = 2

What happened to us? And the following happened. We, in fact, took and ... just threw out the same bases (twos)! Completely thrown out. And, what pleases, hit the bull's-eye!

Yes, indeed, if in the exponential equation on the left and right are the same numbers in any degree, then these numbers can be discarded and simply equate the exponents. Mathematics allows.) And then you can work separately with indicators and solve a much simpler equation. It's great, right?

That's key idea solution of any (yes, exactly any!) exponential equation: via identical transformations it is necessary to ensure that the left and right in the equation are the same base numbers in various degrees. And then you can safely remove the same bases and equate the exponents. And work with a simpler equation.

And now we remember the iron rule: it is possible to remove the same bases if and only if in the equation on the left and on the right the base numbers are in proud loneliness.

What does it mean, in splendid isolation? This means without any neighbors and coefficients. I explain.

For example, in the equation

3 3 x-5 = 3 2 x +1

You can't remove triplets! Why? Because on the left we have not just a lonely three in degree, but work 3 3 x-5 . An extra triple gets in the way: a coefficient, you understand.)

The same can be said about the equation

5 3 x = 5 2 x +5 x

Here, too, all bases are the same - five. But on the right we do not have a single degree of five: there is the sum of the degrees!

In short, we have the right to remove the same bases only when our exponential equation looks like this and only like this:

af (x) = a g (x)

This type of exponential equation is called the simplest. Or scientifically, canonical . And no matter what the twisted equation in front of us may be, one way or another, we will reduce it to such a simple (canonical) form. Or, in some cases, to aggregates equations of this kind. Then our simplest equation can be rewritten in general form as follows:

F(x) = g(x)

And that's it. This will equivalent transformation. At the same time, absolutely any expressions with x can be used as f(x) and g(x). Whatever.

Perhaps a particularly inquisitive student will ask: why on earth do we so easily and simply discard the same bases on the left and right and equate the exponents? Intuition by intuition, but suddenly, in some equation and for some reason this approach turn out to be wrong? Is it always legal to throw the same bases? Unfortunately, for a rigorous mathematical answer to this interest Ask you need to go deep enough and seriously into general theory device and function behavior. And a little more specifically - in the phenomenon strict monotonicity. In particular, the strict monotonicity exponential functiony= a x. Because it exponential function and its properties underlie the solution of exponential equations, yes.) A detailed answer to this question will be given in a separate special lesson devoted to solving complex non-standard equations using the monotonicity of different functions.)

To explain this point in detail now is only to take out the brain of an average schoolchild and scare him ahead of time with a dry and heavy theory. I will not do this.) For our main this moment task - learn to solve exponential equations! The very simplest! Therefore, until we sweat and boldly throw out the same reasons. This is can, take my word for it!) And then we already solve the equivalent equation f (x) = g (x). As a rule, it is simpler than the original exponential.

It is assumed, of course, that people already know how to solve at least , and equations, already without x in indicators.) Who still doesn’t know how, feel free to close this page, walk along the appropriate links and fill in the old gaps. Otherwise, you will have a hard time, yes ...

I am silent about irrational, trigonometric and other brutal equations that can also emerge in the process of eliminating bases. But do not be alarmed, for now we will not consider frank tin in terms of degrees: it's too early. We will train only on the simplest equations.)

Now consider equations that require some additional effort to reduce them to the simplest. To distinguish them, let's call them simple exponential equations. So let's move on to the next level!

Level 1. Simple exponential equations. Recognize degrees! natural indicators.

The key rules in solving any exponential equations are rules for dealing with degrees. Without this knowledge and skills, nothing will work. Alas. So, if there are problems with the degrees, then for a start you are welcome. In addition, we also need . These transformations (as many as two!) are the basis for solving all equations of mathematics in general. And not only showcases. So, whoever forgot, also take a walk on the link: I put them on for a reason.

But only actions with powers and identical transformations are not enough. It also requires personal observation and ingenuity. We need the same grounds, don't we? So we examine the example and look for them in an explicit or disguised form!

For example, this equation:

3 2x – 27x +2 = 0

First look at grounds. They are different! Three and twenty-seven. But it is too early to panic and fall into despair. It's time to remember that

27 = 3 3

Numbers 3 and 27 are relatives in degree! Moreover, relatives.) Therefore, we have every right to write down:

27 x +2 = (3 3) x+2

And now we connect our knowledge about actions with powers(and I warned you!). There is such a very useful formula:

(am) n = a mn

Now if you run it in the course, it generally turns out fine:

27 x +2 = (3 3) x+2 = 3 3(x +2)

The original example now looks like this:

3 2 x – 3 3(x +2) = 0

Great, the bases of the degrees have aligned. What we were striving for. Half the job is done.) And now we launch the basic identity transformation - we transfer 3 3 (x +2) to the right. Nobody canceled the elementary actions of mathematics, yes.) We get:

3 2 x = 3 3(x +2)

What gives us this kind of equation? And the fact that now our equation is reduced to canonical form: standing left and right same numbers(triples) in powers. And both triplets - in splendid isolation. We boldly remove the triplets and get:

2x = 3(x+2)

We solve this and get:

X=-6

That's all there is to it. This is the correct answer.)

And now we comprehend the course of the decision. What saved us in this example? We were saved by the knowledge of the degrees of the triple. How exactly? We identified number 27 encrypted three! This trick (encryption of the same base under different numbers) is one of the most popular in exponential equations! Unless it's the most popular. Yes, and also, by the way. That is why observation and the ability to recognize powers of other numbers in numbers are so important in exponential equations!

Practical advice:

You need to know the powers of popular numbers. In the face!

Of course, anyone can raise two to the seventh power or three to the fifth. Not in my mind, so at least on a draft. But in exponential equations, it is much more often necessary not to raise to a power, but, on the contrary, to find out what number and to what extent is hidden behind the number, say, 128 or 243. And this is already more complicated than simple exponentiation, you see. Feel the difference, as they say!

Since the ability to recognize degrees in the face will be useful not only at this level, but also at the following ones, here is a small task for you:

Determine what powers and what numbers are numbers:

4; 8; 16; 27; 32; 36; 49; 64; 81; 100; 125; 128; 216; 243; 256; 343; 512; 625; 729; 1024.

Answers (scattered, of course):

27 2 ; 2 10 ; 3 6 ; 7 2 ; 2 6 ; 9 2 ; 3 4 ; 4 3 ; 10 2 ; 2 5 ; 3 5 ; 7 3 ; 16 2 ; 2 7 ; 5 3 ; 2 8 ; 6 2 ; 3 3 ; 2 9 ; 2 4 ; 2 2 ; 4 5 ; 25 2 ; 4 4 ; 6 3 ; 8 2 ; 9 3 .

Yes Yes! Do not be surprised that there are more answers than tasks. For example, 2 8 , 4 4 and 16 2 are all 256.

Level 2. Simple exponential equations. Recognize degrees! Negative and fractional exponents.

At this level, we already use our knowledge of degrees to the fullest. Namely, we involve in this fascinating process negative and fractional exponents! Yes Yes! We need to build up power, right?

For example, this terrible equation:

Again, first look at the foundations. The bases are different! And this time not even remotely similar friend on a friend! 5 and 0.04... And to eliminate the bases, the same ones are needed... What to do?

It's OK! In fact, everything is the same, just the connection between the five and 0.04 is visually poorly visible. How do we get out? And let's move on to the number 0.04 to ordinary fraction! And there, you see, everything is formed.)

0,04 = 4/100 = 1/25

Wow! It turns out that 0.04 is 1/25! Well, who would have thought!)

Well, how? Now the connection between the numbers 5 and 1/25 is easier to see? That's what it is...

And now, according to the rules of operations with powers with negative indicator can be written with a firm hand:

That is great. So we got to the same base - five. We now replace the uncomfortable number 0.04 in the equation with 5 -2 and get:

Again, according to the rules of operations with powers, we can now write:

(5 -2) x -1 = 5 -2(x -1)

Just in case, I remind (suddenly, who does not know) that ground rules actions with powers are valid for any indicators! Including for negative ones.) So feel free to take and multiply the indicators (-2) and (x-1) according to the corresponding rule. Our equation gets better and better:

Everything! In addition to the lonely fives in the degrees on the left and right, there is nothing else. The equation is reduced to canonical form. And then - along the knurled track. We remove the fives and equate the indicators:

x 2 –6 x+5=-2(x-1)

The example is almost done. The elementary mathematics of the middle classes remains - we open (correctly!) The brackets and collect everything on the left:

x 2 –6 x+5 = -2 x+2

x 2 –4 x+3 = 0

We solve this and get two roots:

x 1 = 1; x 2 = 3

That's all.)

Now let's think again. AT this example we again had to recognize the same number in varying degrees! Namely, to see the encrypted five in the number 0.04. And this time, in negative degree! How did we do it? On the move - no way. But after the transition from decimal fraction 0.04 to the ordinary fraction 1/25 everything was highlighted! And then the whole decision went like clockwork.)

Therefore, another green practical advice.

If there are decimal fractions in the exponential equation, then we move from decimal fractions to ordinary ones. AT common fractions it's much easier to recognize powers of many popular numbers! After recognition, we move from fractions to powers with negative exponents.

Keep in mind that such a feint in exponential equations occurs very, very often! And the person is not in the subject. He looks, for example, at the numbers 32 and 0.125 and gets upset. It is unknown to him that this is the same deuce, only in varying degrees… But you are already in the subject!)

Solve the equation:

In! It looks like a quiet horror ... However, appearances are deceiving. This is the simplest exponential equation, despite its terrifying appearance. And now I'll show it to you.)

First, we deal with all the numbers sitting in the bases and in the coefficients. They are obviously different, yes. But we still take the risk and try to make them the same! Let's try to get to the same number in different degrees. And, preferably, the number of the smallest possible. So, let's start deciphering!

Well, everything is clear with the four at once - it's 2 2 . So, already something.)

With a fraction of 0.25 - it is not yet clear. Need to check. We use practical advice - go from decimal to ordinary:

0,25 = 25/100 = 1/4

Already much better. For now it is already clearly visible that 1/4 is 2 -2. Great, and the number 0.25 is also akin to a deuce.)

So far so good. But the worst number of all remains - the square root of two! What to do with this pepper? Can it also be represented as a power of two? And who knows...

Well, again we climb into our treasury of knowledge about degrees! This time we additionally connect our knowledge about the roots. From the course of the 9th grade, you and I had to endure that any root, if desired, can always be turned into a degree with a fraction.

Like this:

In our case:

How! It turns out that the square root of two is 2 1/2. That's it!

That's fine! All our uncomfortable numbers actually turned out to be an encrypted deuce.) I do not argue, somewhere very sophisticatedly encrypted. But we also increase our professionalism in solving such ciphers! And then everything is already obvious. We replace the numbers 4, 0.25 and the root of two in our equation with a power of two:

Everything! The bases of all degrees in the example have become the same - two. And now the standard actions with degrees are used:

a ma n = a m + n

a m:a n = a m-n

(am) n = a mn

For the left side you get:

2 -2 (2 2) 5 x -16 = 2 -2+2(5 x -16)

For the right side will be:

And now our evil equation began to look like this:

For those who haven’t figured out how exactly this equation turned out, then the question is not about exponential equations. The question is about actions with powers. I asked urgently to repeat to those who have problems!

Here is the finish line! The canonical form of the exponential equation is obtained! Well, how? Have I convinced you that it's not so scary? ;) We remove the deuces and equate the indicators:

It remains only to solve this linear equation. How? With the help of identical transformations, of course.) Solve what's already there! Multiply both parts by two (to remove the fraction 3/2), move the terms with Xs to the left, without Xs to the right, bring similar ones, count - and you will be happy!

Everything should turn out beautifully:

X=4

Now let's rethink the decision. In this example, we were rescued by the transition from square root to degree with exponent 1/2. Moreover, only such a cunning transformation helped us everywhere to reach same base(two), which saved the day! And, if not for it, then we would have every chance to freeze forever and never cope with this example, yes ...

Therefore, we do not neglect the next practical advice:

If there are roots in the exponential equation, then we go from roots to powers with fractional indicators. Very often, only such a transformation clarifies the further situation.

Of course, negative and fractional powers are already much more difficult. natural degrees. At least in terms of visual perception and, especially, recognition from right to left!

It is clear that directly raising, for example, a two to the power of -3 or a four to the power of -3/2 is not so a big problem. For those who know.)

But go, for example, immediately realize that

0,125 = 2 -3

Or

Here only practice and rich experience rule, yes. And, of course, a clear view, What is a negative and a fractional exponent. As well as - practical advice! Yes, yes, those green.) I hope that they will nevertheless help you to better navigate in all the motley variety of degrees and significantly increase your chances of success! So let's not neglect them. I'm not in vain in green I write sometimes.)

On the other hand, if you become “you” even with such exotic powers as negative and fractional, then your possibilities in solving exponential equations will expand tremendously, and you will already be able to handle almost any type of exponential equations. Well, if not any, then 80 percent of all exponential equations - for sure! Yes, yes, I'm not kidding!

So, our first part of acquaintance with exponential equations has come to its logical conclusion. And, as an in-between workout, I traditionally suggest solving a bit on your own.)

Exercise 1.

So that my words about deciphering the negative and fractional powers not in vain, I propose to play a little game!

Express the number as a power of two:

Answers (in disarray):

Happened? Fine! Then we do a combat mission - we solve the simplest and simple exponential equations!

Task 2.

Solve equations (all answers are a mess!):

5 2x-8 = 25

2 5x-4 – 16x+3 = 0

Answers:

x=16

x 1 = -1; x 2 = 2

x = 5

Happened? Indeed, much easier!

Then we solve the following game:

(2 x +4) x -3 = 0.5 x 4 x -4

35 1-x = 0.2 - x 7 x

Answers:

x 1 = -2; x 2 = 2

x = 0,5

x 1 = 3; x 2 = 5

And these examples of one left? Fine! You are growing! Then here are some more examples for you to snack on:

Answers:

x = 6

x = 13/31

x = -0,75

x 1 = 1; x 2 = 8/3

And is it decided? Well, respect! I take off my hat.) So, the lesson was not in vain, and First level solving exponential equations can be considered successfully mastered. Ahead - next levels and more complex equations! And new techniques and approaches. And non-standard examples. And new surprises.) All this - in the next lesson!

Something didn't work? So, most likely, the problems are in . Or in . Or both at the same time. Here I am powerless. Can in once more offer only one thing - do not be lazy and take a walk through the links.)

To be continued.)