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How to solve equations with powers. What is an exponential equation and how to solve it

At the stage of preparation for the final test, 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 thoroughly master the theory, remember the formulas and understand the principle of solving such equations. Having learned to cope with this type of problem, graduates can count on high scores when passing the Unified State Exam in mathematics.

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When reviewing the materials they have covered, many students are faced with the problem of finding the formulas needed to solve equations. A school textbook is not always at hand, and selecting the necessary information on a 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 the final test. By studying on our website, you will be able to identify gaps in knowledge and pay attention to those tasks that cause the most difficulty.

Shkolkovo teachers collected, systematized and presented everything necessary for successful completion Unified State Exam material in the simplest and most accessible form.

Basic definitions and formulas are presented in the “Theoretical background” section.

To better understand the material, we recommend that you practice completing the assignments. Carefully review the examples presented on this page. exponential equations with the solution to understand the calculation algorithm. After that, proceed to perform tasks in the “Directories” 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.

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What is an exponential equation? Examples.

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

For example, these simple equations:

3 x +1 = 81

5 x + 5 x +2 = 130

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

Or even these monsters:

2 sin x = 0.5

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

Generally speaking, not all and not always even pure exponential equations can be solved clearly. But among all the rich variety of exponential equations, there are certain types that can and should be solved. It is these types of equations that we will consider. And we’ll definitely solve the examples.) So let’s get comfortable and off we go! As in computer shooters, our journey will take place through levels.) From elementary to simple, from simple to intermediate and from intermediate to complex. Along the way, a secret level will also await you - techniques and methods for solving non-standard examples. Those that you won’t read about in most school textbooks... Well, and at the end, of course, the final boss awaits you in the form of homework.)

Level 0. What is the simplest exponential equation? Solving simple exponential equations.

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

2 x = 2 2

Even without any theories, according to simple logic and common sense It’s clear that x = 2. There’s no other way, right? No other meaning of X is suitable... And now let’s turn our attention to record of decision this cool exponential equation:

2 x = 2 2

X = 2

What happened to us? And the following happened. We actually took it and... just threw it away identical grounds(twos)! Completely thrown out. And, the good news is, we hit the bull’s eye!

Yes, indeed, if in an exponential equation there are left and right the same numbers in any powers, then these numbers can be discarded and simply equate the exponents. Mathematics allows.) And then you can work separately with the indicators and solve a much simpler equation. Great, right?

Here is the key idea for solving any (yes, exactly any!) exponential equation: by using identity transformations it is necessary to ensure that the left and right in the equation are the same base numbers in various powers. And then you can safely remove the same bases and equate the exponents. And work with a simpler equation.

Now let’s remember the iron rule: it is possible to remove identical bases if and only if the numbers on the left and right of the equation have base numbers in proud loneliness.

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

For example, in Eq.

3 3 x-5 = 3 2 x +1

Threes cannot be removed! Why? Because on the left we have not just a lonely three to the degree, but work 3·3 x-5 . An extra three interferes: the coefficient, you understand.)

The same can be said about the equation

5 3 x = 5 2 x +5 x

Here, too, all the bases are the same - five. But on the right we don’t have a single power of five: there is a sum of powers!

In short, we have the right to remove identical 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 convoluted equation we have in front of us, we will, one way or another, reduce it to precisely this simplest (canonical) form. Or, in some cases, to totality equations of this type. Then our simplest equation can be written as general view rewrite it like this:

F(x) = g(x)

That's all. This would be an equivalent conversion. In this case, f(x) and g(x) can be absolutely any expressions with an x. Whatever.

Perhaps a particularly inquisitive student will wonder: why on earth do we so easily and simply discard the same bases on the left and right and equate the exponents? Intuition is intuition, but what if, in some equation and for some reason, this approach turns out to be incorrect? Is it always legal to throw out the same grounds? Unfortunately, for a rigorous mathematical answer to this interest Ask you need to dive quite deeply and seriously into the general theory of the structure and behavior of functions. And a little more specifically - in the phenomenon strict monotony. In particular, strict monotony exponential function y= a x. Since it is the exponential function and its properties that underlie the solution of exponential equations, yes.) A detailed answer to this question will be given in a separate special lesson dedicated to solving complex non-standard equations using the monotonicity of different functions.)

Explaining this point in detail now would only blow the minds of the average student and scare him away ahead of time with a dry and heavy theory. I won’t do this.) Because our main this moment task - learn to solve exponential equations! The simplest ones! Therefore, let’s not worry yet and boldly throw out the same reasons. This Can, take my word for it!) And then we solve the equivalent equation f(x) = g(x). As a rule, simpler than the original exponential.

It is assumed, of course, that people already know how to solve at least , and equations, without x’s in exponents.) For those who still don’t know how, feel free to close this page, follow the relevant links and fill in the old gaps. Otherwise you will have a hard time, yes...

I'm not talking about irrational, trigonometric and other brutal equations that can also emerge in the process of eliminating the foundations. But don’t be alarmed, we won’t consider outright cruelty in terms of degrees for now: it’s too early. We will train only on the simplest equations.)

Now let's look at equations that require some additional effort to reduce them to the simplest. For the sake of distinction, let's call them simple exponential equations. So, let's move to the next level!

Level 1. Simple exponential equations. Let's recognize the 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 first you are welcome. In addition, we will also need . These transformations (two of them!) are the basis for solving all mathematical equations in general. And not only demonstrative ones. So, whoever forgot, also take a look at the link: I don’t just put them there.

But operations with powers and identity transformations alone are not enough. Personal observation and ingenuity are also required. We need the same reasons, don't we? So we examine the example and look for them in an explicit or disguised form!

For example, this equation:

3 2 x – 27 x +2 = 0

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

27 = 3 3

Numbers 3 and 27 are relatives by degree! And close ones.) Therefore, we have every right to write:

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

Now let’s connect our knowledge about actions with degrees(and I warned you!). There is a very useful formula there:

(a m) n = a mn

If you now put it into action, it works out great:

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 leveled out. That's what we wanted. Half the battle is done.) Now we launch the basic identity transformation - move 3 3(x +2) to the right. No one has canceled the elementary operations of mathematics, yes.) We get:

3 2 x = 3 3(x +2)

What does this type of equation give us? And the fact that now our equation is reduced to canonical form: left and right stand same numbers(threes) in degrees. Moreover, both three are in splendid isolation. Feel free to remove the triples and get:

2x = 3(x+2)

We solve this and get:

X = -6

That's it. This is the correct answer.)

Now let’s think about the solution. What saved us in this example? Knowledge of the powers of three saved us. How exactly? We identified number 27 contains an 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 in the same way, by the way. This 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 face!

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

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

Determine what powers and what numbers the numbers are:

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

Answers (randomly, 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! Don't 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. Let's recognize the degrees! Negative and fractional indicators.

At this level we are already using our knowledge of degrees to the fullest. Namely, we involve negative and fractional indicators in this fascinating process! Yes Yes! We need to increase our power, right?

For example, this terrible equation:

Again, the first glance is at the foundations. The reasons 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, it’s just that the connection between the five and 0.04 is visually poorly visible. How can we get out? Let's move on to the number 0.04 to common fraction! And then, you see, everything will work out.)

0,04 = 4/100 = 1/25

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

So how? Is it now easier to see the connection between the numbers 5 and 1/25? That's it...

And now according to the rules of actions with degrees with negative indicator You can write with a steady hand:

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

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

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

Just in case, I remind you (in case anyone doesn’t know) that the basic rules for dealing with degrees 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 appropriate rule. Our equation gets better and better:

All! Apart from lonely fives, there is nothing else in the powers on the left and right. 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 solved. All that's left is elementary middle school math - 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. IN in this example we again had to recognize the same number to different degrees! Namely, to see an encrypted five in the number 0.04. And this time - in negative degree! How did we do this? Right off the bat - no way. But after the transition from decimal 0.04 to the common fraction 1/25 and that’s it! And then the whole decision went like clockwork.)

Therefore, another green practical advice.

If an exponential equation contains decimal fractions, then we move from decimal fractions to ordinary fractions. It's much easier to recognize powers of many popular numbers in fractions! After recognition, we move from fractions to powers with negative exponents.

Keep in mind that this trick occurs very, very often in exponential equations! But the person is not in the subject. He looks, for example, at the numbers 32 and 0.125 and gets upset. Unbeknownst to him, this is one and the same deuce, only in different degrees...But you’re already on topic!)

Solve the equation:

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

First, let’s look at all the numbers in the bases and coefficients. They are, of course, different, yes. But we will still take a risk and try to make them identical! Let's try to get to the same number in different powers. Moreover, preferably, the numbers are as small as possible. So, let's start decoding!

Well, with the four everything is immediately clear - it’s 2 2. Okay, that's something already.)

With a fraction of 0.25 – it’s still unclear. Need to check. Let's use practical advice - move from a decimal fraction to an ordinary fraction:

0,25 = 25/100 = 1/4

Much better already. Because now it is clearly visible that 1/4 is 2 -2. Great, and the number 0.25 is also akin to two.)

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

Well, let's dive into our treasury of knowledge about degrees again! This time we additionally connect our knowledge about roots. From the 9th grade course, you and I should have learned that any root, if desired, can always be turned into a degree with a fractional indicator.

Like this:

In our case:

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

That's fine! All our inconvenient numbers actually turned out to be an encrypted two.) I don’t argue, somewhere very sophisticatedly encrypted. But we are also improving our professionalism in solving such ciphers! And then everything is already obvious. In our equation we replace the numbers 4, 0.25 and the root of two by powers of two:

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

a ma n = a m + n

a m:a n = a m-n

(a m) 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 it will be:

And now our evil equation looks like this:

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

Here is the finish line! The canonical form of the exponential equation has been obtained! So how? Have I convinced you that everything is not so scary? ;) We remove the twos and equate the indicators:

All that's left to do is solve it linear equation. How? With the help of identical transformations, of course.) Decide what’s going on! Multiply both sides by two (to remove the fraction 3/2), move the terms with X's to the left, without X's to the right, bring similar ones, count - and you will be happy!

Everything should turn out beautifully:

X=4

Now let’s think about the solution again. In this example, we were helped by the transition from square root To degree with exponent 1/2. Moreover, only such a cunning transformation helped us reach the same base (two) everywhere, which saved the situation! 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 an exponential equation contains roots, then we move from roots to powers with fractional exponents. Very often only such a transformation clarifies the further situation.

Of course, negative and fractional powers are already much more complex than natural powers. At least from the point of view of visual perception and, especially, recognition from right to left!

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

But go, for example, immediately realize that

0,125 = 2 -3

Here, only practice and rich experience rule, yes. And, of course, a clear idea, What is a negative and fractional degree? And also practical advice! Yes, yes, those same ones green.) I hope that they will still help you better navigate the entire diverse variety of degrees and significantly increase your chances of success! So let's not neglect them. I'm not in vain green I write sometimes.)

But if you get to know each other even with such exotic powers as negative and fractional ones, then your capabilities in solving exponential equations will expand enormously, and you will 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 joking!

So, our first part of our introduction to exponential equations has come to its logical conclusion. And, as an intermediate workout, I traditionally suggest doing a little self-reflection.)

Exercise 1.

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

Express numbers as powers of two:

Answers (in disarray):

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

Task 2.

Solve the equations (all answers are a mess!):

5 2x-8 = 25

2 5x-4 – 16 x+3 = 0

Answers:

x = 16

x 1 = -1; x 2 = 2

x = 5

Happened? Indeed, it’s much simpler!

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

Did something go wrong? This means that most likely the problems are in . Or in . Or both at once. I'm powerless here. I can once again suggest only one thing - don’t be lazy and follow the links.)

To be continued.)

Go to the youtube channel of our website to stay up to date with all the new video lessons.

First, let's remember the basic formulas of powers and their properties.

Product of a number a occurs on itself n times, we can write this expression as a a … a=a n

1. a 0 = 1 (a ≠ 0)

3. a n a m = a n + m

4. (a n) m = a nm

5. a n b n = (ab) n

7. a n / a m = a n - m

Power or exponential equations– these are equations in which the variables are in powers (or exponents), and the base is a number.

Examples of exponential equations:

In this example, the number 6 is the base; it is always at the bottom, and the variable x degree or indicator.

Let us give more examples of exponential equations.
2 x *5=10
16 x - 4 x - 6=0

Now let's look at how exponential equations are solved?

Let's take a simple equation:

2 x = 2 3

This example can be solved even in your head. It can be seen that x=3. After all, in order for the left and right sides to be equal, you need to put the number 3 instead of x.
Now let’s see how to formalize this decision:

2 x = 2 3
x = 3

In order to solve such an equation, we removed identical grounds(that is, twos) and wrote down what was left, these are degrees. We got the answer we were looking for.

Now let's summarize our decision.

Algorithm for solving the exponential equation:
1. Need to check the same whether the equation has bases on the right and left. If the reasons are not the same, we are looking for options to solve this example.
2. After the bases become the same, equate degrees and solve the resulting new equation.

Now let's look at a few examples:

Let's start with something simple.

The bases on the left and right sides are equal to the number 2, which means we can discard the base and equate their degrees.

x+2=4 The simplest equation is obtained.
x=4 – 2
x=2
Answer: x=2

IN following example It can be seen that the bases are different: 3 and 9.

3 3x - 9 x+8 = 0

First, move the nine to the right side, we get:

Now you need to make the same bases. We know that 9=3 2. Let's use the power formula (a n) m = a nm.

3 3x = (3 2) x+8

We get 9 x+8 =(3 2) x+8 =3 2x+16

3 3x = 3 2x+16 Now it is clear that on the left and right sides the bases are the same and equal to three, which means we can discard them and equate the degrees.

3x=2x+16 we get the simplest equation
3x - 2x=16
x=16
Answer: x=16.

Let's look at the following example:

2 2x+4 - 10 4 x = 2 4

First of all, we look at the bases, bases two and four. And we need them to be the same. We transform the four using the formula (a n) m = a nm.

4 x = (2 2) x = 2 2x

And we also use one formula a n a m = a n + m:

2 2x+4 = 2 2x 2 4

Add to the equation:

2 2x 2 4 - 10 2 2x = 24

We gave an example for the same reasons. But other numbers 10 and 24 bother us. What to do with them? If you look closely you can see that on the left side we have 2 2x repeated, here is the answer - we can put 2 2x out of brackets:

2 2x (2 4 - 10) = 24

Let's calculate the expression in brackets:

2 4 — 10 = 16 — 10 = 6

We divide the entire equation by 6:

Let's imagine 4=2 2:

2 2x = 2 2 bases are the same, we discard them and equate the degrees.
2x = 2 is the simplest equation. Divide it by 2 and we get
x = 1
Answer: x = 1.

Let's solve the equation:

9 x – 12*3 x +27= 0

Let's convert:
9 x = (3 2) x = 3 2x

We get the equation:
3 2x - 12 3 x +27 = 0

Our bases are the same, equal to three. In this example, you can see that the first three has a degree twice (2x) than the second (just x). In this case, you can solve replacement method. We replace the number with the smallest degree:

Then 3 2x = (3 x) 2 = t 2

We replace all x powers in the equation with t:

t 2 - 12t+27 = 0
We get a quadratic equation. Solving through the discriminant, we get:
D=144-108=36
t 1 = 9
t2 = 3

Returning to the variable x.

Take t 1:
t 1 = 9 = 3 x

That is,

3 x = 9
3 x = 3 2
x 1 = 2

One root was found. We are looking for the second one from t 2:
t 2 = 3 = 3 x
3 x = 3 1
x 2 = 1
Answer: x 1 = 2; x 2 = 1.

On the website you can ask any questions you may have in the HELP DECIDE section, we will definitely answer you.

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Lecture: “Methods for solving exponential equations.”

1 . Exponential equations.

Equations containing unknowns in exponents are called exponential equations. The simplest of them is the equation ax = b, where a > 0, a ≠ 1.

1) At b< 0 и b = 0 это уравнение, согласно свойству 1 показательной функции, не имеет решения.

2) For b > 0, using the monotonicity of the function and the root theorem, the equation has a unique root. In order to find it, b must be represented in the form b = aс, аx = bс ó x = c or x = logab.

Exponential equations through algebraic transformations lead to standard equation which are solved using the following methods:

1) method of reduction to one base;

2) assessment method;

3) graphic method;

4) method of introducing new variables;

5) factorization method;

6) exponential – power equations;

7) demonstrative with a parameter.

2 . Method of reduction to one base.

The method is based on the following property of degrees: if two degrees are equal and their bases are equal, then their exponents are equal, i.e., one must try to reduce the equation to the form

Examples. Solve the equation:

1 . 3x = 81;

Let's imagine right side equations in the form 81 = 34 and write the equation equivalent to the original 3 x = 34; x = 4. Answer: 4.

2. https://pandia.ru/text/80/142/images/image004_8.png" width="52" height="49">and let's move on to the equation for exponents 3x+1 = 3 – 5x; 8x = 4; x = 0.5 Answer: 0.5.

3. https://pandia.ru/text/80/142/images/image006_8.png" width="105" height="47">

Note that the numbers 0.2, 0.04, √5 and 25 represent powers of 5. Let's take advantage of this and transform the original equation as follows:

, whence 5-x-1 = 5-2x-2 ó - x – 1 = - 2x – 2, from which we find the solution x = -1. Answer: -1.

5. 3x = 5. By definition of logarithm, x = log35. Answer: log35.

6. 62x+4 = 33x. 2x+8.

Let's rewrite the equation in the form 32x+4.22x+4 = 32x.2x+8, i.e..png" width="181" height="49 src="> Hence x – 4 =0, x = 4. Answer: 4.

7 . 2∙3x+1 - 6∙3x-2 - 3x = 9. Using the properties of powers, we write the equation in the form 6∙3x - 2∙3x – 3x = 9 then 3∙3x = 9, 3x+1 = 32, i.e. i.e. x+1 = 2, x =1. Answer: 1.

Problem bank No. 1.

Solve the equation:

Test No. 1.

	1) 0 2) 4 3) -2 4) -4
A2 32x-8 = √3.	1)17/4 2) 17 3) 13/2 4) -17/4
A3	1) 3;1 2) -3;-1 3) 0;2 4) no roots
	1) 7;1 2) no roots 3) -7;1 4) -1;-7
A5	1) 0;2; 2) 0;2;3 3) 0 4) -2;-3;0
A6	1) -1 2) 0 3) 2 4) 1

Test No. 2

A1	1) 3 2) -1;3 3) -1;-3 4) 3;-1
A2	1) 14/3 2) -14/3 3) -17 4) 11
A3	1) 2;-1 2) no roots 3) 0 4) -2;1
A4	1) -4 2) 2 3) -2 4) -4;2
A5	1) 3 2) -3;1 3) -1 4) -1;3

3 Evaluation method.

Root theorem: if the function f(x) increases (decreases) on the interval I, the number a is any value taken by f on this interval, then the equation f(x) = a has a single root on the interval I.

When solving equations using the estimation method, this theorem and the monotonicity properties of the function are used.

Examples. Solve equations: 1. 4x = 5 – x.

Solution. Let's rewrite the equation as 4x +x = 5.

1. if x = 1, then 41+1 = 5, 5 = 5 is true, which means 1 is the root of the equation.

Function f(x) = 4x – increases on R, and g(x) = x – increases on R => h(x)= f(x)+g(x) increases on R, as the sum of increasing functions, then x = 1 is the only root of the equation 4x = 5 – x. Answer: 1.

Solution. Let's rewrite the equation in the form .

1. if x = -1, then , 3 = 3 is true, which means x = -1 is the root of the equation.

2. prove that he is the only one.

3. Function f(x) = - decreases on R, and g(x) = - x – decreases on R=> h(x) = f(x)+g(x) – decreases on R, as the sum of decreasing functions . This means, according to the root theorem, x = -1 is the only root of the equation. Answer: -1.

Problem bank No. 2. Solve the equation

a) 4x + 1 =6 – x;

c) 2x – 2 =1 – x;

4. Method of introducing new variables.

The method is described in paragraph 2.1. The introduction of a new variable (substitution) is usually carried out after transformations (simplification) of the terms of the equation. Let's look at examples.

Examples. R Solve the equation: 1. .

Let's rewrite the equation differently: https://pandia.ru/text/80/142/images/image030_0.png" width="128" height="48 src="> i.e..png" width="210" height ="45">

Solution. Let's rewrite the equation differently:

Let's designate https://pandia.ru/text/80/142/images/image035_0.png" width="245" height="57"> - not suitable.

t = 4 => https://pandia.ru/text/80/142/images/image037_0.png" width="268" height="51"> - irrational equation. We note that

The solution to the equation is x = 2.5 ≤ 4, which means 2.5 is the root of the equation. Answer: 2.5.

Solution. Let's rewrite the equation in the form and divide both sides by 56x+6 ≠ 0. We get the equation

2x2-6x-7 = 2x2-6x-8 +1 = 2(x2-3x-4)+1, t..png" width="118" height="56">

The roots of the quadratic equation are t1 = 1 and t2<0, т. е..png" width="200" height="24">.

Solution . Let's rewrite the equation in the form

and note that it is a homogeneous equation of the second degree.

Divide the equation by 42x, we get

Let's replace https://pandia.ru/text/80/142/images/image049_0.png" width="16" height="41 src="> .

Answer: 0; 0.5.

Problem bank No. 3. Solve the equation

Test No. 3 with a choice of answers. Minimum level.

A1	1) -0.2;2 2) log52 3) –log52 4) 2
A2 0.52x – 3 0.5x +2 = 0.	1) 2;1 2) -1;0 3) no roots 4) 0
	1) 0 2) 1; -1/3 3) 1 4) 5
A4 52x-5x - 600 = 0.	1) -24;25 2) -24,5; 25,5 3) 25 4) 2
	1) no roots 2) 2;4 3) 3 4) -1;2

Test No. 4 with a choice of answers. General level.

A1	1) 2;1 2) ½;0 3)2;0 4) 0
A2 2x – (0.5)2x – (0.5)x + 1 = 0	1) -1;1 2) 0 3) -1;0;1 4) 1
	1) 64 2) -14 3) 3 4) 8
	1)-1 2) 1 3) -1;1 4) 0
A5	1) 0 2) 1 3) 0;1 4) no roots

5. Factorization method.

1. Solve the equation: 5x+1 - 5x-1 = 24.

Solution..png" width="169" height="69"> , from where

2. 6x + 6x+1 = 2x + 2x+1 + 2x+2.

Solution. Let's put 6x out of brackets on the left side of the equation, and 2x on the right side. We get the equation 6x(1+6) = 2x(1+2+4) ó 6x = 2x.

Since 2x >0 for all x, we can divide both sides of this equation by 2x without fear of losing solutions. We get 3x = 1ó x = 0.

Solution. Let's solve the equation using the factorization method.

Let us select the square of the binomial

4. https://pandia.ru/text/80/142/images/image067_0.png" width="500" height="181">

x = -2 is the root of the equation.

Equation x + 1 = 0 " style="border-collapse:collapse;border:none">

A1 5x-1 +5x -5x+1 =-19.

1) 1 2) 95/4 3) 0 4) -1

A2 3x+1 +3x-1 =270.

1) 2 2) -4 3) 0 4) 4

A3 32x + 32x+1 -108 = 0. x=1.5

1) 0,2 2) 1,5 3) -1,5 4) 3

1) 1 2) -3 3) -1 4) 0

A5 2x -2x-4 = 15. x=4

1) -4 2) 4 3) -4;4 4) 2

Test No. 6 General level.

A1 (22x-1)(24x+22x+1)=7.	1) ½ 2) 2 3) -1;3 4) 0.2
A2	1) 2.5 2) 3;4 3) log43/2 4) 0
A3 2x-1-3x=3x-1-2x+2.	1) 2 2) -1 3) 3 4) -3
A4	1) 1,5 2) 3 3) 1 4) -4
A5	1) 2 2) -2 3) 5 4) 0

6. Exponential – power equations.

Adjacent to exponential equations are the so-called exponential-power equations, i.e., equations of the form (f(x))g(x) = (f(x))h(x).

If it is known that f(x)>0 and f(x) ≠ 1, then the equation, like the exponential one, is solved by equating the exponents g(x) = f(x).

If the condition does not exclude the possibility of f(x)=0 and f(x)=1, then we have to consider these cases when solving an exponential equation.

1..png" width="182" height="116 src=">

Solution. x2 +2x-8 – makes sense for any x, since it is a polynomial, which means the equation is equivalent to the totality

https://pandia.ru/text/80/142/images/image078_0.png" width="137" height="35">

7. Exponential equations with parameters.

1. For what values of the parameter p does equation 4 (5 – 3)2 +4p2–3p = 0 (1) have a unique solution?

Solution. Let us introduce the replacement 2x = t, t > 0, then equation (1) will take the form t2 – (5p – 3)t + 4p2 – 3p = 0. (2)

Discriminant of equation (2) D = (5p – 3)2 – 4(4p2 – 3p) = 9(p – 1)2.

Equation (1) has a unique solution if equation (2) has one positive root. This is possible in the following cases.

1. If D = 0, that is, p = 1, then equation (2) will take the form t2 – 2t + 1 = 0, hence t = 1, therefore, equation (1) has a unique solution x = 0.

2. If p1, then 9(p – 1)2 > 0, then equation (2) has two different roots t1 = p, t2 = 4p – 3. The conditions of the problem are satisfied by a set of systems

Substituting t1 and t2 into the systems, we have

https://pandia.ru/text/80/142/images/image084_0.png" alt="no35_11" width="375" height="54"> в зависимости от параметра a?!}

Solution. Let then equation (3) will take the form t2 – 6t – a = 0. (4)

Let us find the values of the parameter a for which at least one root of equation (4) satisfies the condition t > 0.

Let us introduce the function f(t) = t2 – 6t – a. The following cases are possible.

https://pandia.ru/text/80/142/images/image087.png" alt="http://1september.ru/ru/mat/2002/35/no35_14.gif" align="left" width="215" height="73 src=">где t0 - абсцисса вершины параболы и D - дискриминант квадратного трехчлена f(t);!}

https://pandia.ru/text/80/142/images/image089.png" alt="http://1september.ru/ru/mat/2002/35/no35_16.gif" align="left" width="60" height="51 src=">!}

Case 2. Equation (4) has a unique positive solution if

D = 0, if a = – 9, then equation (4) will take the form (t – 3)2 = 0, t = 3, x = – 1.

Case 3. Equation (4) has two roots, but one of them does not satisfy the inequality t > 0. This is possible if

https://pandia.ru/text/80/142/images/image092.png" alt="no35_17" width="267" height="63">!}

Thus, for a 0, equation (4) has a single positive root . Then equation (3) has a unique solution

When a< – 9 уравнение (3) корней не имеет.

if a< – 9, то корней нет; если – 9 < a < 0, то
if a = – 9, then x = – 1;

if a  0, then

Let us compare the methods for solving equations (1) and (3). Note that when solving equation (1) was reduced to a quadratic equation, the discriminant of which is a perfect square; Thus, the roots of equation (2) were immediately calculated using the formula for the roots of a quadratic equation, and then conclusions were drawn regarding these roots. Equation (3) has been reduced to a quadratic equation (4), the discriminant of which is not a perfect square, therefore, when solving equation (3), it is advisable to use theorems on the location of the roots of a quadratic trinomial and a graphical model. Note that equation (4) can be solved using Vieta's theorem.

Let's solve more complex equations.

Problem 3: Solve the equation

Solution. ODZ: x1, x2.

Let's introduce a replacement. Let 2x = t, t > 0, then as a result of transformations the equation will take the form t2 + 2t – 13 – a = 0. (*) Let us find the values of a for which at least one root of the equation (*) satisfies the condition t > 0.

https://pandia.ru/text/80/142/images/image098.png" alt="http://1september.ru/ru/mat/2002/35/no35_23.gif" align="left" width="71" height="68 src=">где t0 - абсцисса вершины f(t) = t2 + 2t – 13 – a, D - дискриминант квадратного трехчлена f(t).!}

https://pandia.ru/text/80/142/images/image100.png" alt="http://1september.ru/ru/mat/2002/35/no35_25.gif" align="left" width="360" height="32 src=">!}

https://pandia.ru/text/80/142/images/image102.png" alt="http://1september.ru/ru/mat/2002/35/no35_27.gif" align="left" width="218" height="42 src=">!}

Answer: if a > – 13, a  11, a  5, then if a – 13,

a = 11, a = 5, then there are no roots.

Bibliography.

1. Guzeev foundations of educational technology.

2. Guzeev technology: from reception to philosophy.

M. “School Director” No. 4, 1996

3. Guzeev and organizational forms of training.

4. Guzeev and the practice of integral educational technology.

M. " Public education", 2001

5. Guzeev from the forms of a lesson - seminar.

Mathematics at school No. 2, 1987 pp. 9 – 11.

6. Seleuko educational technologies.

M. “Public Education”, 1998

7. Episheva schoolchildren to study mathematics.

M. "Enlightenment", 1990

8. Ivanova prepare lessons - workshops.

Mathematics at school No. 6, 1990 p. 37 – 40.

9. Smirnov’s model of teaching mathematics.

Mathematics at school No. 1, 1997 p. 32 – 36.

10. Tarasenko ways of organizing practical work.

Mathematics at school No. 1, 1993 p. 27 – 28.

11. About one of the types of individual work.

Mathematics at school No. 2, 1994, pp. 63 – 64.

12. Khazankin Creative skills schoolchildren.

Mathematics at school No. 2, 1989 p. 10.

13. Scanavi. Publisher, 1997

14. and others. Algebra and the beginnings of analysis. Didactic materials For

15. Krivonogov tasks in mathematics.

M. “First of September”, 2002

16. Cherkasov. Handbook for high school students and

entering universities. “A S T - press school”, 2002

17. Zhevnyak for those entering universities.

Minsk and Russian Federation “Review”, 1996

18. Written D. We are preparing for the exam in mathematics. M. Rolf, 1999

19. etc. Learning to solve equations and inequalities.

M. "Intellect - Center", 2003

20. etc. Educational and training materials for preparing for the EGE.

M. "Intelligence - Center", 2003 and 2004.

21 and others. CMM options. Testing Center of the Ministry of Defense of the Russian Federation, 2002, 2003.

22. Goldberg equations. "Quantum" No. 3, 1971

23. Volovich M. How to successfully teach mathematics.

Mathematics, 1997 No. 3.

24 Okunev for the lesson, children! M. Education, 1988

25. Yakimanskaya – oriented learning At school.

26. Liimets work in class. M. Knowledge, 1975

Solving exponential equations. Examples.

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

What's happened exponential equation? This is an equation in which the unknowns (x's) and expressions with them are in indicators some degrees. And only there! It is important.

There you are examples of exponential equations:

3 x 2 x = 8 x+3

Note! In the bases of degrees (below) - only numbers. IN indicators degrees (above) - a wide variety of expressions with an X. If, suddenly, an X appears in the equation somewhere other than an indicator, for example:

this will already be an equation of mixed type. Such equations do not have clear rules for solving them. We will not consider them for now. Here we will deal with solving exponential equations in its purest form.

In fact, even pure exponential equations are not always solved clearly. But there are certain types of exponential equations that can and should be solved. These are the types we will consider.

Solving simple exponential equations.

First, let's solve something very basic. For example:

Even without any theories, by simple selection it is clear that x = 2. Nothing more, right!? No other value of X works. Now let's look at the solution to this tricky exponential equation:

What have we done? We, in fact, simply threw out the same bases (triples). Completely thrown out. And, the good news is, we hit the nail on the head!

Indeed, if in an exponential equation there are left and right the same numbers in any powers, these numbers can be removed and the exponents can be equalized. Mathematics allows. It remains to solve a much simpler equation. Great, right?)

However, let us remember firmly: You can remove bases only when the base numbers on the left and right are in splendid isolation! Without any neighbors and coefficients. Let's say in the equations:

2 x +2 x+1 = 2 3, or

twos cannot be removed!

Well, we have mastered the most important thing. How to move from evil exponential expressions to simpler equations.

"Those are the times!" - you say. “Who would give such a primitive lesson on tests and exams!?”

I have to agree. Nobody will. But now you know where to aim when solving tricky examples. It must be brought to the form where the same base number is on the left and right. Then everything will be easier. Actually, this is a classic of mathematics. We take the original example and transform it to the desired one us mind. According to the rules of mathematics, of course.

Let's look at examples that require some additional effort to reduce them to the simplest. Let's call them simple exponential equations.

Solving simple exponential equations. Examples.

When solving exponential equations, the main rules are actions with degrees. Without knowledge of these actions nothing will work.

To actions with degrees, one must add personal observation and ingenuity. Do we need the same base numbers? So we look for them in the example in explicit or encrypted form.

Let's see how this is done in practice?

Let us be given an example:

2 2x - 8 x+1 = 0

The first keen glance is at grounds. They... They are different! Two and eight. But it’s too early to become discouraged. It's time to remember that

Two and eight are relatives in degree.) It is quite possible to write:

8 x+1 = (2 3) x+1

If we recall the formula from operations with degrees:

(a n) m = a nm ,

this works out great:

8 x+1 = (2 3) x+1 = 2 3(x+1)

The original example began to look like this:

2 2x - 2 3(x+1) = 0

We transfer 2 3 (x+1) to the right (no one has canceled the elementary operations of mathematics!), we get:

2 2x = 2 3(x+1)

That's practically all. Removing the bases:

We solve this monster and get

This is the correct answer.

In this example, knowing the powers of two helped us out. We identified in eight there is an encrypted two. This technique (encryption common grounds under different numbers) is a very popular technique in exponential equations! Yes, and in logarithms too. You must be able to recognize powers of other numbers in numbers. This is extremely important for solving exponential equations.

The fact is that raising any number to any power is not a problem. Multiply, even on paper, and that’s it. For example, anyone can raise 3 to the fifth power. 243 will work out if you know the multiplication table.) But in exponential equations, much more often it is not necessary to raise to a power, but vice versa... Find out what number to what degree is hidden behind the number 243, or, say, 343... No calculator will help you here.

You need to know the powers of some numbers by sight, right... Let's practice?

Determine what powers and what numbers the numbers are:

2; 8; 16; 27; 32; 64; 81; 100; 125; 128; 216; 243; 256; 343; 512; 625; 729, 1024.

Answers (in a mess, of course!):

5 4 ; 2 10 ; 7 3 ; 3 5 ; 2 7 ; 10 2 ; 2 6 ; 3 3 ; 2 3 ; 2 1 ; 3 6 ; 2 9 ; 2 8 ; 6 3 ; 5 3 ; 3 4 ; 2 5 ; 4 4 ; 4 2 ; 2 3 ; 9 3 ; 4 5 ; 8 2 ; 4 3 ; 8 3 .

If you look closely, you can see a strange fact. There are significantly more answers than tasks! Well, it happens... For example, 2 6, 4 3, 8 2 - that's all 64.

Let us assume that you have taken note of the information about familiarity with numbers.) Let me also remind you that to solve exponential equations we use all stock of mathematical knowledge. Including those from junior and middle classes. You didn’t go straight to high school, right?)

For example, when solving exponential equations, putting the common factor out of brackets often helps (hello to 7th grade!). Let's look at an example:

3 2x+4 -11 9 x = 210

And again, the first glance is at the foundations! The bases of the degrees are different... Three and nine. But we want them to be the same. Well, in this case the desire is completely fulfilled!) Because:

9 x = (3 2) x = 3 2x

Using the same rules for dealing with degrees:

3 2x+4 = 3 2x ·3 4

That’s great, you can write it down:

3 2x 3 4 - 11 3 2x = 210

We gave an example for the same reasons. So, what is next!? You can't throw out threes... Dead end?

Not at all. Remember the most universal and powerful decision rule everyone math tasks:

If you don’t know what you need, do what you can!

Look, everything will work out).

What's in this exponential equation Can do? Yes, on the left side it just begs to be taken out of brackets! The overall multiplier of 3 2x clearly hints at this. Let's try, and then we'll see:

3 2x (3 4 - 11) = 210

3 4 - 11 = 81 - 11 = 70

The example keeps getting better and better!

We remember that to eliminate grounds we need a pure degree, without any coefficients. The number 70 bothers us. So we divide both sides of the equation by 70, we get:

Oops! Everything got better!

This is the final answer.

It happens, however, that taxiing on the same basis is achieved, but their elimination is not possible. This happens in other types of exponential equations. Let's master this type.

Replacing a variable in solving exponential equations. Examples.

Let's solve the equation:

4 x - 3 2 x +2 = 0

First - as usual. Let's move on to one base. To a deuce.

4 x = (2 2) x = 2 2x

We get the equation:

2 2x - 3 2 x +2 = 0

And this is where we hang out. The previous techniques will not work, no matter how you look at it. We'll have to pull out another powerful and universal method from our arsenal. It's called variable replacement.

The essence of the method is surprisingly simple. Instead of one complex icon (in our case - 2 x) we write another, simpler one (for example - t). Such a seemingly meaningless replacement leads to amazing results!) Everything just becomes clear and understandable!

So let

Then 2 2x = 2 x2 = (2 x) 2 = t 2

In our equation we replace all powers with x's by t:

Well, is it dawning on you?) Quadratic equations Have you forgotten yet? Solving through the discriminant, we get:

The main thing here is not to stop, as happens... This is not the answer yet, we need x, not t. Let's return to the X's, i.e. we make a reverse replacement. First for t 1:

That is,

One root was found. We are looking for the second one from t 2:

Hm... 2 x on the left, 1 on the right... Problem? Not at all! It is enough to remember (from operations with powers, yes...) that a unit is any number to the zero power. Any. Whatever is needed, we will install it. We need a two. Means:

That's it now. We got 2 roots:

This is the answer.

At solving exponential equations at the end sometimes you end up with some kind of awkward expression. Type:

Seven cannot be converted to two through a simple power. They are not relatives... How can we be? Someone may be confused... But the person who read on this site the topic “What is a logarithm?” , just smiles sparingly and writes down with a firm hand the absolutely correct answer:

There cannot be such an answer in tasks “B” on the Unified State Examination. There a specific number is required. But in tasks “C” it’s easy.

This lesson provides examples of solving the most common exponential equations. Let's highlight the main points.

Practical advice:

1. First of all, we look at grounds degrees. We are wondering if it is possible to make them identical. Let's try to do this by actively using actions with degrees. Don't forget that numbers without x's can also be converted to powers!

2. We try to bring the exponential equation to the form when on the left and on the right there are the same numbers in any powers. We use actions with degrees And factorization. What can be counted in numbers, we count.

3. If the second tip doesn’t work, try using variable replacement. The result may be an equation that can be easily solved. Most often - square. Or fractional, which also reduces to square.

4. To successfully solve exponential equations, you need to know the powers of some numbers by sight.

As usual, at the end of the lesson you are invited to decide a little.) On your own. From simple to complex.

Solve exponential equations:

More difficult:

2 x+3 - 2 x+2 - 2 x = 48

9 x - 8 3 x = 9

2 x - 2 0.5x+1 - 8 = 0

Find the product of roots:

2 3's + 2 x = 9

Happened?

Well then the most complicated example(decided, however, in the mind...):

7 0.13x + 13 0.7x+1 + 2 0.5x+1 = -3

What's more interesting? Then here's a bad example for you. Quite drawn to increased difficulty. Let me hint that in this example, what saves you is ingenuity and the most universal rule for solving all mathematical problems.)

2 5x-1 3 3x-1 5 2x-1 = 720 x

A simpler example, for relaxation):

9 2 x - 4 3 x = 0

And for dessert. Find the sum of the roots of the equation:

x 3 x - 9x + 7 3 x - 63 = 0

Yes Yes! This is a mixed type equation! Which we did not consider in this lesson. Why consider them, they need to be solved!) This lesson is quite enough to solve the equation. Well, you need ingenuity... And may seventh grade help you (this is a hint!).

Answers (in disarray, separated by semicolons):

1; 2; 3; 4; there are no solutions; 2; -2; -5; 4; 0.

Is everything successful? Great.

There is a problem? No problem! Special Section 555 solves all these exponential equations with detailed explanations. What, why, and why. And, of course, there is additional valuable information on working with all sorts of exponential equations. Not just these ones.)

One last fun question to consider. In this lesson we worked with exponential equations. Why didn’t I say a word about ODZ here? In equations, this is a very important thing, by the way...