Belgorod State University

DEPARTMENT algebra, number theory and geometry

Work theme: Exponential power equations and inequalities.

Graduate work student of the Faculty of Physics and Mathematics

Scientific adviser:

______________________________

Reviewer: _______________________________

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Belgorod. 2006

Introduction			3
Subject I.	Analysis of literature on the research topic.
Subject II.	Functions and their properties used in solving exponential equations and inequalities.
	I.1.	Power function and its properties.
	I.2.	Exponential function and its properties.
Subject III.	Solving exponential power equations, algorithm and examples.
Subject IV.	Solving exponential inequalities, solution plan and examples.
Subject V.	Experience in conducting classes with schoolchildren on the topic: “Solving exponential equations and inequalities.”
V. 1.	Educational material.
V. 2.	Problems for independent solution.
Conclusion.	Conclusions and offers.
Bibliography.
Applications

Introduction.

“...the joy of seeing and understanding...”

A. Einstein.

In this work, I tried to convey my experience as a mathematics teacher, to convey at least to some extent my attitude towards its teaching - a human endeavor in which mathematical science, pedagogy, didactics, psychology, and even philosophy are surprisingly intertwined.

I had the opportunity to work with kids and graduates, with children at the extremes of intellectual development: those who were registered with a psychiatrist and who were really interested in mathematics

I had the opportunity to solve many methodological problems. I will try to talk about those that I managed to solve. But even more failed, and even in those that seem to have been resolved, new questions arise.

But even more important than the experience itself are the teacher’s reflections and doubts: why is it exactly like this, this experience?

And the summer is different now, and the development of education has become more interesting. “Under the Jupiters” today is not a search for a mythical optimal system of teaching “everyone and everything,” but the child himself. But then - of necessity - the teacher.

IN school course algebra and beginning of analysis, grades 10 - 11, when passing the Unified State Exam for the course high school and on entrance exams to universities there are equations and inequalities containing an unknown in the base and exponents - these are exponential equations and inequalities.

They receive little attention at school; there are practically no assignments on this topic in textbooks. However, mastering the methodology for solving them, it seems to me, is very useful: it increases the mental and creative abilities of students, and completely new horizons open up before us. When solving problems, students acquire first skills research work, their mathematical culture is enriched, their abilities to logical thinking. Schoolchildren develop such personality qualities as determination, goal-setting, independence, which will be useful to them in later life. And also there is repetition, expansion and deep assimilation of educational material.

I started working on this topic for my thesis by writing my coursework. In the course of which I deeply studied and analyzed the mathematical literature on this topic, I identified the most suitable method for solving exponential equations and inequalities.

It lies in the fact that in addition to the generally accepted approach when solving exponential equations (the base is taken greater than 0) and when solving the same inequalities (the base is taken greater than 1 or greater than 0, but less than 1), cases are also considered when the bases are negative, equal 0 and 1.

Analysis of written exam papers students shows that the lack of coverage of the issue of negative value argument of the exponential function in school textbooks causes them a number of difficulties and leads to errors. And they also have problems at the stage of systematizing the results obtained, where, due to the transition to an equation - a consequence or an inequality - a consequence, extraneous roots may appear. In order to eliminate errors, we use a test using the original equation or inequality and an algorithm for solving exponential equations, or a plan for solving exponential inequalities.

In order for students to successfully pass final and entrance exams, I believe it is necessary to pay more attention to solving exponential equations and inequalities in classes, or additionally in electives and clubs.

Thus subject , my thesis is defined as follows: “Exponential power equations and inequalities.”

Goals of this work are:

1. Analyze the literature on this topic.

2. Give full analysis solving exponential power equations and inequalities.

3. Provide a sufficient number of examples of various types on this topic.

4. Check in class, elective and club classes how the proposed methods for solving exponential equations and inequalities will be perceived. Give appropriate recommendations for studying this topic.

Subject Our research is to develop a methodology for solving exponential equations and inequalities.

The purpose and subject of the study required solving the following problems:

1. Study the literature on the topic: “Exponential power equations and inequalities.”

2. Master the techniques for solving exponential equations and inequalities.

3. Select training material and develop a system of exercises different levels on the topic: “Solving exponential equations and inequalities.”

During the thesis research, more than 20 papers were analyzed on the use of various methods for solving exponential equations and inequalities. From here we get.

Thesis plan:

Introduction.

Chapter I. Analysis of literature on the research topic.

Chapter II. Functions and their properties used in solving exponential equations and inequalities.

II.1. Power function and its properties.

II.2. Exponential function and its properties.

Chapter III. Solving exponential power equations, algorithm and examples.

Chapter IV. Solving exponential inequalities, solution plan and examples.

Chapter V. Experience of conducting classes with schoolchildren on this topic.

1.Training material.

2.Tasks for independent solution.

Conclusion. Conclusions and offers.

List of used literature.

Chapter I analyzes the literature

First level

Exponential equations. Comprehensive Guide (2019)

Hello! Today we will discuss with you how to solve equations that can be either elementary (and I hope that after reading this article, almost all of them will be so for you), and those that are usually given “for filling”. Apparently to finally fall asleep. But I will try to do everything possible so that now you don’t get into trouble when faced with this type of equations. I won't beat around the bush anymore, I'll open it right away little secret: today we will study exponential equations.

Before moving on to analyzing ways to solve them, I will immediately outline for you a range of questions (quite small) that you should repeat before rushing to attack this topic. So, for best results, please repeat:

1. Properties and
2. Solution and equations

Repeated? Amazing! Then it will not be difficult for you to notice that the root of the equation is a number. Do you understand exactly how I did it? Is it true? Then let's continue. Now answer my question, what is equal to the third power? You're absolutely right: . What power of two is eight? That's right - the third one! Because. Well, now let's try to solve the following problem: Let me multiply the number by itself once and get the result. The question is, how many times did I multiply by myself? You can of course check this directly:

\begin(align) & 2=2 \\ & 2\cdot 2=4 \\ & 2\cdot 2\cdot 2=8 \\ & 2\cdot 2\cdot 2\cdot 2=16 \\ \end( align)

Then you can conclude that I multiplied by myself times. How else can you check this? Here's how: directly by definition of degree: . But, you must admit, if I asked how many times two needs to be multiplied by itself to get, say, you would tell me: I won’t fool myself and multiply by itself until I’m blue in the face. And he would be absolutely right. Because how can you write down all the steps briefly(and brevity is the sister of talent)

where - these are the same ones "times", when you multiply by itself.

I think that you know (and if you don’t know, urgently, very urgently repeat the degrees!) that then my problem will be written in the form:

How can you reasonably conclude that:

So, unnoticed, I wrote down the simplest exponential equation:

And I even found him root. Don't you think that everything is completely trivial? I think exactly the same. Here's another example for you:

But what to do? After all, it cannot be written as a power of a (reasonable) number. Let's not despair and note that both of these numbers are perfectly expressed through the power of the same number. Which one? Right: . Then the original equation is transformed to the form:

Where, as you already understood, . Let's not delay any longer and write it down definition:

In our case: .

These equations are solved by reducing them to the form:

followed by solving the equation

In fact, in the previous example we did just that: we got the following: And we solved the simplest equation.

It seems like nothing complicated, right? Let's practice on the simplest ones first examples:

We again see that the right and left sides of the equation need to be represented as powers of one number. True, on the left this has already been done, but on the right there is a number. But it’s okay, because my equation will miraculously transform into this:

What did I have to use here? What rule? Rule of "degrees within degrees" which reads:

What if:

Before answering this question, let’s fill out the following table:

It is easy for us to notice that the less, the less value, but nevertheless, all these values ​​are greater than zero. AND IT WILL ALWAYS BE SO!!! The same property is true FOR ANY BASIS WITH ANY INDICATOR!! (for any and). Then what can we conclude about the equation? Here's what it is: it has no roots! Just like any equation has no roots. Now let's practice and Let's solve simple examples:

Let's check:

1. Here nothing will be required of you except knowledge of the properties of degrees (which, by the way, I asked you to repeat!) As a rule, everything leads to the smallest base: , . Then the original equation will be equivalent to the following: All I need is to use the properties of powers: When multiplying numbers with the same bases, the powers are added, and when dividing, they are subtracted. Then I will get: Well, now with a clear conscience I will move from the exponential equation to the linear one: \begin(align)
& 2x+1+2(x+2)-3x=5 \\
& 2x+1+2x+4-3x=5 \\
&x=0. \\
\end(align)

2. In the second example, we need to be more careful: the trouble is that on the left side we can’t possibly represent the same number as a power. In this case it is sometimes useful represent numbers as a product of powers with different bases, but the same exponents:

The left side of the equation will look like: What did this give us? Here's what: Numbers with different bases but the same exponents can be multiplied.In this case, the bases are multiplied, but the indicator does not change:

In my situation this will give:

\begin(align)
& 4\cdot ((64)^(x))((25)^(x))=6400,\\
& 4\cdot (((64\cdot 25))^(x))=6400,\\
& ((1600)^(x))=\frac(6400)(4), \\
& ((1600)^(x))=1600, \\
&x=1. \\
\end(align)

Not bad, right?

3. I don’t like it when, unnecessarily, I have two terms on one side of the equation and none on the other (sometimes, of course, this is justified, but this is not the case now). I’ll move the minus term to the right:

Now, as before, I’ll write everything in terms of powers of three:

I add the degrees on the left and get an equivalent equation

You can easily find its root:

4. As in example three, the minus term has a place on the right side!

On my left, almost everything is fine, except for what? Yes, the “wrong degree” of the two is bothering me. But I can easily fix this by writing: . Eureka - on the left all the bases are different, but all the degrees are the same! Let's multiply immediately!

Here again everything is clear: (if you don’t understand how magically I got the last equality, take a break for a minute, take a breath and read the properties of the degree again very carefully. Who said you can skip a degree with a negative score? Well, that's what I'm saying, no one). Now I will get:

\begin(align)
& ((2)^(4\left((x) -9 \right)))=((2)^(-1)) \\
& 4((x) -9)=-1 \\
& x=\frac(35)(4). \\
\end(align)

Here are some problems for you to practice, to which I will only give the answers (but in a “mixed” form). Solve them, check them, and you and I will continue our research!

Ready? Answers like these ones:

1. any number

Okay, okay, I was joking! Here are some sketches of solutions (some very brief!)

Don't you think it's no coincidence that one fraction on the left is the other one "inverted"? It would be a sin not to take advantage of this:

This rule is very often used when solving exponential equations, remember it well!

Then the original equation will become like this:

Having decided this quadratic equation, you will get these roots:

2. Another solution: dividing both sides of the equation by the expression on the left (or right). Divide by what is on the right, then I get:

Where (why?!)

3. I don’t even want to repeat myself, everything has already been “chewed” so much.

4. equivalent to a quadratic equation, roots

5. You need to use the formula given in the first problem, then you will get that:

The equation has turned into a trivial identity that is true for any. Then the answer is any real number.

Well, now you’ve practiced solving simple exponential equations. Now I want to give you a few life examples that will help you understand why they are needed in principle. Here I will give two examples. One of them is quite everyday, but the other is more likely to be of scientific rather than practical interest.

Example 1 (mercantile) Let you have rubles, but you want to turn it into rubles. The bank offers you to take this money from you at an annual rate with monthly capitalization of interest (monthly accrual). The question is, how many months do you need to open a deposit for to reach the required final amount? Quite a mundane task, isn’t it? Nevertheless, its solution is associated with the construction of the corresponding exponential equation: Let - the initial amount, - the final amount, - interest rate per period, - the number of periods. Then:

In our case (if the rate is annual, then it is calculated per month). Why is it divided by? If you don’t know the answer to this question, remember the topic “”! Then we get this equation:

This exponential equation can only be solved using a calculator (its appearance hints at this, and this requires knowledge of logarithms, which we will get acquainted with a little later), which I will do: ... Thus, in order to receive a million, we will need to make a deposit for a month (not very quickly, right?).

Example 2 (rather scientific). Despite his certain “isolation”, I recommend that you pay attention to him: he regularly “slips into the Unified State Examination!! (the problem is taken from the “real” version) During the decay of a radioactive isotope, its mass decreases according to the law, where (mg) is the initial mass of the isotope, (min.) is the time elapsed from the initial moment, (min.) is the half-life. At the initial moment of time, the mass of the isotope is mg. Its half-life is min. After how many minutes will the mass of the isotope be equal to mg? It’s okay: we just take and substitute all the data into the formula proposed to us:

Let's divide both parts by, "in the hope" that on the left we will get something digestible:

Well, we are very lucky! It’s on the left, then let’s move on to the equivalent equation:

Where is min.

As you can see, exponential equations have very real applications in practice. Now I want to show you another (simple) way to solve exponential equations, which is based on taking the common factor out of brackets and then grouping the terms. Don't be scared by my words, you already came across this method in 7th grade when you studied polynomials. For example, if you needed to factor the expression:

Let's group: the first and third terms, as well as the second and fourth. It is clear that the first and third are the difference of squares:

and the second and fourth have a common factor of three:

Then the original expression is equivalent to this:

Where to derive the common factor is no longer difficult:

Hence,

This is roughly what we will do when solving exponential equations: look for “commonality” among the terms and take it out of brackets, and then - come what may, I believe that we will be lucky =)) For example:

On the right is far from being a power of seven (I checked!) And on the left - it’s a little better, you can, of course, “chop off” the factor a from the second from the first term, and then deal with what you got, but let’s be more prudent with you. I don't want to deal with the fractions that inevitably form when "selecting" , so shouldn't I rather take it out? Then I won’t have any fractions: as they say, the wolves are fed and the sheep are safe:

Calculate the expression in brackets. Magically, magically, it turns out that (surprisingly, although what else should we expect?).

Then we reduce both sides of the equation by this factor. We get: , from.

Here's a more complicated example (quite a bit, really):

What a problem! We don't have one common ground here! It's not entirely clear what to do now. Let’s do what we can: first, move the “fours” to one side, and the “fives” to the other:

Now let's take out the "general" on the left and right:

So what now? What is the benefit of such a stupid group? At first glance it is not visible at all, but let's look deeper:

Well, now we’ll make sure that on the left we only have the expression c, and on the right - everything else. How do we do this? Here's how: Divide both sides of the equation first by (so we get rid of the exponent on the right), and then divide both sides by (so we get rid of the numeric factor on the left). Finally we get:

Incredible! On the left we have an expression, and on the right we have a simple expression. Then we immediately conclude that

Here's another example for you to reinforce:

I will give his brief solution (without bothering myself much with explanations), try to understand all the “subtleties” of the solution yourself.

Now for the final consolidation of the material covered. Try to solve the following problems yourself. I will just give brief recommendations and tips for solving them:

1. Let's take the common factor out of brackets: Where:
2. Let's present the first expression in the form: , divide both sides by and get that
3. , then the original equation is transformed to the form: Well, now a hint - look for where you and I have already solved this equation!
4. Imagine how, how, ah, well, then divide both sides by, so you get the simplest exponential equation.
5. Bring it out of the brackets.
6. Bring it out of the brackets.

EXPONENTARY EQUATIONS. AVERAGE LEVEL

I assume that after reading the first article, which talked about what are exponential equations and how to solve them, you have mastered the necessary minimum knowledge necessary to solve the simplest examples.

Now I will look at another method for solving exponential equations, this is

“method of introducing a new variable” (or replacement). He solves most “difficult” problems on the topic of exponential equations (and not only equations). This method is one of the most frequently used in practice. First, I recommend that you familiarize yourself with the topic.

As you already understood from the name, the essence of this method is to introduce such a change of variable that your exponential equation will miraculously transform into one that you can easily solve. All that remains for you after solving this very “simplified equation” is to make a “reverse replacement”: that is, return from the replaced to the replaced. Let's illustrate what we just said with a very simple example:

Example 1:

This equation is solved using a “simple substitution,” as mathematicians disparagingly call it. In fact, the replacement here is the most obvious. One has only to see that

Then the original equation will turn into this:

If we additionally imagine how, then it is absolutely clear what needs to be replaced: of course, . What then becomes the original equation? Here's what:

You can easily find its roots on your own: . What should we do now? It's time to return to the original variable. What did I forget to mention? Namely: when replacing a certain degree with a new variable (that is, when replacing a type), I will be interested in only positive roots! You yourself can easily answer why. Thus, you and I are not interested, but the second root is quite suitable for us:

Then where from.

Answer:

As you can see, in the previous example, a replacement was just asking for our hands. Unfortunately, this is not always the case. However, let’s not go straight to the sad stuff, but let’s practice with one more example with a fairly simple replacement

Example 2.

It is clear that most likely we will have to make a replacement (this is the smallest of the powers included in our equation), but before introducing a replacement, our equation needs to be “prepared” for it, namely: , . Then you can replace, as a result I get the following expression:

Oh horror: a cubic equation with absolutely terrible formulas for solving it (well, speaking in general view). But let’s not despair right away, but let’s think about what we should do. I'll suggest cheating: we know that to get a “beautiful” answer, we need to get it in the form of some power of three (why would that be, eh?). Let's try to guess at least one root of our equation (I'll start guessing with powers of three).

First guess. Not a root. Alas and ah...

.
The left side is equal.
Right part: !
Eat! Guessed the first root. Now things will get easier!

Do you know about the “corner” division scheme? Of course you do, you use it when you divide one number by another. But few people know that the same can be done with polynomials. There is one wonderful theorem:

Applying to my situation, this tells me that it is divisible without remainder by. How is division carried out? That's how:

I look at which monomial I should multiply by to get Clearly, then:

I subtract the resulting expression from, I get:

Now, what do I need to multiply by to get? It is clear that on, then I will get:

and again subtract the resulting expression from the remaining one:

Well, the last step is to multiply by and subtract from the remaining expression:

Hurray, division is over! What have we accumulated in private? By itself: .

Then we got the following expansion of the original polynomial:

Let's solve the second equation:

It has roots:

Then the original equation:

has three roots:

We will, of course, discard the last root, since it is less than zero. And the first two after reverse replacement will give us two roots:

Answer: ..

With this example, I did not at all want to scare you; rather, my goal was to show that although we had a fairly simple replacement, it nevertheless led to a rather complex equation, the solution of which required some special skills from us. Well, no one is immune from this. But the replacement in in this case was pretty obvious.

Here's an example with a slightly less obvious replacement:

It is not at all clear what we should do: the problem is that in our equation there are two different bases and one base cannot be obtained from the other by raising it to any (reasonable, naturally) power. However, what do we see? Both bases differ only in sign, and their product is the difference of squares equal to one:

Definition:

Thus, the numbers that are the bases in our example are conjugate.

In this case, the smart step would be multiply both sides of the equation by the conjugate number.

For example, on, then the left side of the equation will become equal to, and the right. If we make a substitution, then our original equation will become like this:

its roots, then, and remembering that, we get that.

Answer: , .

As a rule, the replacement method is sufficient to solve most “school” exponential equations. The following tasks are taken from the Unified State Examination C1 ( increased level difficulties). You are already literate enough to solve these examples on your own. I will only give the required replacement.

1. Solve the equation:
2. Find the roots of the equation:
3. Solve the equation: . Find all the roots of this equation that belong to the segment:

And now some brief explanations and answers:

1. Here it is enough for us to note that... Then the original equation will be equivalent to this: This equation can be solved by replacing Do the further calculations yourself. In the end, your task will be reduced to solving simple trigonometric problems (depending on sine or cosine). We will look at solutions to similar examples in other sections.
2. Here you can even do without substitution: just move the subtrahend to the right and represent both bases through powers of two: , and then go straight to the quadratic equation.
3. The third equation is also solved quite standardly: let’s imagine how. Then, replacing, we get a quadratic equation: then,

   You already know what a logarithm is, right? No? Then read the topic urgently!

   The first root obviously does not belong to the segment, but the second one is unclear! But we will find out very soon! Since, then (this is a property of the logarithm!) Let’s compare:

   Subtract from both sides, then we get:

   The left side can be represented as:

   multiply both sides by:

   can be multiplied by, then

   Then compare:

   since then:

   Then the second root belongs to the required interval

   Answer:

As you see, selection of roots of exponential equations requires a fairly deep knowledge of the properties of logarithms, so I advise you to be as careful as possible when solving exponential equations. As you understand, in mathematics everything is interconnected! As my math teacher said: “mathematics, like history, cannot be read overnight.”

As a rule, all The difficulty in solving problems C1 is precisely the selection of the roots of the equation. Let's practice with one more example:

It is clear that the equation itself is solved quite simply. By making a substitution, we reduce our original equation to the following:

First let's look at the first root. Let's compare and: since, then. (property of a logarithmic function, at). Then it is clear that the first root does not belong to our interval. Now the second root: . It is clear that (since the function at is increasing). It remains to compare and...

since, then, at the same time. This way I can “drive a peg” between the and. This peg is a number. The first expression is less and the second is greater. Then the second expression is greater than the first and the root belongs to the interval.

Answer: .

Finally, let's look at another example of an equation where the substitution is quite non-standard:

Let's start right away with what can be done, and what - in principle, can be done, but it is better not to do it. You can imagine everything through the powers of three, two and six. Where it leads? It won’t lead to anything: a jumble of degrees, some of which will be quite difficult to get rid of. What then is needed? Let's note that a And what will this give us? And the fact that we can reduce the solution of this example to the solution of a fairly simple exponential equation! First, let's rewrite our equation as:

Now let's divide both sides of the resulting equation by:

Eureka! Now we can replace, we get:

Well, now it’s your turn to solve exemplary problems, and I’ll only give them brief comments so that you don't go astray the right path! Good luck!

1. The most difficult! It’s so hard to see a replacement here! But nevertheless, this example can be completely solved using discharge full square . To solve it, it is enough to note that:

Then here's your replacement:

(Please note that here during our replacement we cannot discard the negative root!!! Why do you think?)

Now to solve the example you only have to solve two equations:

Both of them can be solved by a “standard replacement” (but the second one in one example!)

2. Notice that and make a replacement.

3. Decompose the number into coprime factors and simplify the resulting expression.

4. Divide the numerator and denominator of the fraction by (or, if you prefer) and make the substitution or.

5. Notice that the numbers and are conjugate.

EXPONENTARY EQUATIONS. ADVANCED LEVEL

In addition, let's look at another way - solving exponential equations using the logarithm method. I can’t say that solving exponential equations using this method is very popular, but in some cases only it can lead us to the right decision our equation. It is especially often used to solve the so-called “ mixed equations": that is, those where functions of different types occur.

For example, an equation of the form:

in the general case, it can only be solved by taking logarithms of both sides (for example, to the base), in which the original equation will turn into the following:

Let's look at the following example:

It is clear that according to the ODZ of the logarithmic function, we are only interested. However, this follows not only from the ODZ of the logarithm, but for one more reason. I think it won’t be difficult for you to guess which one it is.

Let's take the logarithm of both sides of our equation to the base:

As you can see, taking the logarithm of our original equation quickly led us to the correct (and beautiful!) answer. Let's practice with one more example:

There’s nothing wrong here either: let’s take the logarithm of both sides of the equation to the base, then we get:

Let's make a replacement:

However, we missed something! Did you notice where I made a mistake? After all, then:

which does not satisfy the requirement (think where it came from!)

Answer:

Try to write down the solution to the exponential equations below:

Now compare your decision with this:

1. Let’s logarithm both sides to the base, taking into account that:

(the second root is not suitable for us due to replacement)

2. Logarithm to the base:

Let us transform the resulting expression to the following form:

EXPONENTARY EQUATIONS. BRIEF DESCRIPTION AND BASIC FORMULAS

Exponential equation

Equation of the form:

called the simplest exponential equation.

Properties of degrees

Approaches to solution

• Reduction to the same basis
• Reduction to the same exponent
• Variable replacement
• Simplifying the expression and applying one of the above.

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.

Get ready for exam testing with Shkolkovo!

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 have collected, systematized and presented everything necessary for successful passing 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 of exponential equations with solutions presented on this page to understand the calculation algorithm. After that, proceed to perform tasks in the “Directories” section. You can start with the easiest problems 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 “Favorites”. This way you can quickly find them and discuss the solution with your teacher.

To successfully pass the Unified State Exam, study on the Shkolkovo portal every day!

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 in this lesson we will not consider them. 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, a 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 base numbers on the left and right of the equation are 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 interest Ask you need to dive quite deeply and seriously into general theory device and function behavior. And a little more specifically - in the phenomenon strict monotony. In particular, strict monotony exponential functiony= a x. Because exactly 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 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 schoolchild 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 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.) 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 (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, 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 in person 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 basic 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 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. IN ordinary 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 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! By the look - 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. So, 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! Received canonical view exponential equation! 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 everywhere same base(two), 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 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

Or

Here, only practice and rich experience rule, yes. And, of course, a clear idea, What is a negative and fractional degree? And - 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 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 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 in Once again I can only suggest one thing - don’t be lazy and follow the links.)

To be continued.)

The use of equations is widespread in our lives. They are used in many calculations, construction of structures and even sports. Man used equations in ancient times, and since then their use has only increased. Power or exponential equations are equations in which the variables are in powers and the base is a number. For example:

Solving an exponential equation comes down to 2 fairly simple steps:

1. You need to check whether the bases of the equation on the right and left are the same. If the reasons are not the same, we look for options to solve this example.

2. After the bases become the same, we equate the degrees and solve the resulting new equation.

Suppose we are given an exponential equation of the following form:

Start solution given equation costs from the analysis of the basis. The bases are different - 2 and 4, but to solve we need them to be the same, so we transform 4 using the following formula -\[ (a^n)^m = a^(nm):\]

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. A free online solver will allow you to solve the equation online any complexity in 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.