Exponential power equations. Lecture: “Methods for solving exponential equations

1º. Exponential equations are called equations containing a variable in an exponent.

Solution exponential equations based on the property of a degree: two powers with the same base are equal if and only if their exponents are equal.

2º. Basic methods for solving exponential equations:

1) the simplest equation has a solution;

2) an equation of the form logarithmic to the base a reduce to form;

3) an equation of the form is equivalent to the equation ;

4) equation of the form is equivalent to the equation.

5) an equation of the form is reduced through substitution to an equation, and then a set of simple exponential equations is solved;

6) equation with reciprocals by substitution they reduce to an equation, and then solve a set of equations;

7) equations homogeneous with respect to a g(x) And b g(x) given that kind through replacement they are reduced to an equation, and then a set of equations is solved.

Classification of exponential equations.

1. Equations solved by going to one base.

Example 18. Solve the equation .

Solution: Let's take advantage of the fact that all bases of powers are powers of the number 5: .

2. Equations solved by passing to one exponent.

These equations are solved by transforming the original equation to the form , which is reduced to its simplest using the property of proportion.

Example 19. Solve the equation:

3. Equations solved by taking the common factor out of brackets.

If each exponent in an equation differs from the other by a certain number, then the equations are solved by putting the exponent with the smallest exponent out of brackets.

Example 20. Solve the equation.

Solution: Let’s take the degree with the smallest exponent out of brackets on the left side of the equation:

Example 21. Solve the equation

Solution: Let's group separately on the left side of the equation the terms containing powers with base 4, on the right side - with base 3, then put the powers with the smallest exponent out of brackets:

4. Equations that reduce to quadratic (or cubic) equations.

The following equations are reduced to a quadratic equation for the new variable y:

a) the type of substitution, in this case;

b) the type of substitution , and .

Example 22. Solve the equation .

Solution: Let's make a change of variable and solve quadratic equation:

.

Answer: 0; 1.

5. Equations that are homogeneous with respect to exponential functions.

An equation of the form is homogeneous equation second degree relative to unknowns a x And b x. Such equations are reduced by first dividing both sides by and then substituting them into quadratic equations.

Example 23. Solve the equation.

Solution: Divide both sides of the equation by:

Putting , we get a quadratic equation with roots .

Now the problem comes down to solving a set of equations . From the first equation we find that . The second equation has no roots, since for any value x.

Answer: -1/2.

6. Rational equations with respect to exponential functions.

Example 24. Solve the equation.

Solution: Divide the numerator and denominator of the fraction by 3 x and instead of two we get one exponential function:

7. Equations of the form .

Such equations with a set of admissible values ​​(APV), determined by the condition, by taking the logarithm of both sides of the equation are reduced to an equivalent equation, which in turn are equivalent to a set of two equations or.

Example 25. Solve the equation: .

.

Didactic material.

Solve the equations:

1. ; 2. ; 3. ;

4. ; 5. ; 6. ;

9. ; 10. ; 11. ;

14. ; 15. ;

16. ; 17. ;

18. ; 19. ;

20. ; 21. ;

22. ; 23. ;

24. ; 25. .

26. Find the product of the roots of the equation .

27. Find the sum of the roots of the equation .

Find the meaning of the expression:

28. , where x 0- root of the equation ;

29. , where x 0– whole root of the equation .

Solve the equation:

31. ; 32. .

Answers: 10; 2. -2/9; 3. 1/36; 4. 0, 0.5; 50; 6.0; 7. -2; 8.2; 9. 1, 3; 10. 8; 11.5; 12.1; 13. ¼; 14.2; 15. -2, -1; 16. -2, 1; 17.0; 18.1; 19.0; 20. -1, 0; 21. -2, 2; 22. -2, 2; 23.4; 24. -1, 2; 25. -2, -1, 3; 26. -0.3; 27.3; 28.11; 29.54; 30. -1, 0, 2, 3; 31. ; 32. .

Topic No. 8.

Exponential inequalities.

1º. An inequality containing a variable in the exponent is called exponential inequality.

2º. Solution exponential inequalities type is based on the following statements:

if , then the inequality is equivalent to ;

if , then the inequality is equivalent to .

When solving exponential inequalities, the same techniques are used as when solving exponential equations.

Example 26. Solve inequality (method of transition to one base).

Solution: Because , then the given inequality can be written as: . Since , then this inequality is equivalent to the inequality .

Solving the last inequality, we get .

Example 27. Solve the inequality: ( by taking the common factor out of brackets).

Solution: Let's take out of brackets on the left side of the inequality , on the right side of the inequality and divide both sides of the inequality by (-2), changing the sign of the inequality to the opposite:

Since , then when moving to inequality of indicators, the sign of inequality again changes to the opposite. We get. Thus, the set of all solutions to this inequality is the interval.

Example 28. Solve inequality ( by introducing a new variable).

Solution: Let . Then this inequality will take the form: or , whose solution is the interval .

From here. Since the function increases, then .

Didactic material.

Specify the set of solutions to the inequality:

1. ; 2. ; 3. ;

6. At what values x Do the points on the function graph lie below the straight line?

7. At what values x Do the points on the graph of the function lie at least as far as the straight line?

Solve the inequality:

8. ; 9. ; 10. ;

13. Specify the largest integer solution to the inequality .

14. Find the product of the largest integer and the smallest integer solutions to the inequality .

Solve the inequality:

15. ; 16. ; 17. ;

18. ; 19. ; 20. ;

21. ; 22. ; 23. ;

24. ; 25. ; 26. .

Find the domain of the function:

27. ; 28. .

29. Find the set of argument values ​​for which the values ​​of each function are greater than 3:

And .

Answers: 11.3; 12.3; 13. -3; 14.1; 15. (0; 0.5); 16. ; 17. (-1; 0)U(3; 4); 18. [-2; 2]; 19. (0; +∞); 20. (0; 1); 21. (3; +∞); 22. (-∞; 0)U(0.5; +∞); 23. (0; 1); 24. (-1; 1); 25. (0; 2]; 26. (3; 3.5)U (4; +∞); 27. (-∞; 3)U(5); 28.

We add to the original equation:

Let's take it out of brackets \

Let's express \

Since the degrees are the same, we discard them:

Answer: \

Where can I solve an exponential equation using an online solver?

You can solve the equation on our website https://site. The free online solver will allow you to solve online equations of any complexity in a matter of seconds. All you need to do is simply enter your data into the solver. You can also watch video instructions and learn how to solve the equation on our website. And if you still have questions, you can ask them in our VKontakte group http://vk.com/pocketteacher. Join our group, we are always happy to help you.

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. The key word 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, by simple logic and common sense it is clear that x = 2. There is 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... simply threw out the same bases (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: using identical transformations, it is necessary to ensure that the left and right sides of 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 rewritten in general form 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 interesting question, 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 functiony= 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 task at the moment is 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 at the moment 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.) And 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: on the left and right there are the same numbers (threes) in powers. 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 (encoding 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, on the contrary, 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 by sight will be useful not only at this level, but also at the next ones, here’s 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 they are not even remotely similar to each other! 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 as an ordinary 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 this example, we again had to recognize the same number in 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 moving from the decimal fraction 0.04 to the common fraction 1/25, everything became clear! 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 two, only in different degrees... But you’re already in the know!)

Solve the equation:

In! It looks like quiet horror... However, appearances are deceiving. This is the simplest exponential equation, despite its intimidating 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 remains is to solve this 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 of -3 or four to the power of -3/2 is not such a big problem. For those in the 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 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. It’s not for nothing that I sometimes write in green.)

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 do not go in vain, I suggest playing 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.) This means that the lesson was not in vain, and the initial level of solving exponential equations can be considered successfully mastered. Next levels and more complex equations are ahead! 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.)