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 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 be an equation 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. We require same numbers-grounds? 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 of 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 of on the same grounds. 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, does it dawn on you?) Have you forgotten the quadratic equations 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:

From seven to two through simple degree does not work. 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 smile sparingly and write down with a steady hand 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 tempting for 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...

If you like this site...

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

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

You can get acquainted with functions and derivatives.

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, but 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, to get best result, 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 smaller, the smaller the 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 now is not such a case). 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 I magically 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 that you can skip a degree with a negative exponent? Well, here I am about the same thing as 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, which 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, - the interest rate for the 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 simple 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 of the “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 this case was quite 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 foundation cannot be obtained from another by raising it to any (reasonable, naturally) degree. 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 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 demonstration problems, and I will give only brief comments to them so that you don’t go astray! 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.

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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 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 the following example you can see 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 you can see that in the left and right side 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.

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

Solving exponential equations is based on the property of powers: 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 the 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 acceptable values(ODZ), 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 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: Since , 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. (a)=a^(\frac( 1)(n))\) we obtain that \(\sqrt(3^3)=((3^3))^(\frac(1)(2))\). Next, using the property of degree \((a^b)^c=a^(bc)\), we obtain \(((3^3))^(\frac(1)(2))=3^(3 \ cdot \frac(1)(2))=3^(\frac(3)(2))\).

\(3^(\frac(3)(2))\cdot 3^(x-1)=(\frac(1)(3))^(2x)\)

We also know that \(a^b·a^c=a^(b+c)\). Applying this to the left side, we get: \(3^(\frac(3)(2))·3^(x-1)=3^(\frac(3)(2)+ x-1)=3^ (1.5 + x-1)=3^(x+0.5)\).

\(3^(x+0.5)=(\frac(1)(3))^(2x)\)

Now remember that: \(a^(-n)=\frac(1)(a^n)\). This formula can also be used in the opposite direction: \(\frac(1)(a^n) =a^(-n)\). Then \(\frac(1)(3)=\frac(1)(3^1) =3^(-1)\).

\(3^(x+0.5)=(3^(-1))^(2x)\)

Applying the property \((a^b)^c=a^(bc)\) to the right side, we obtain: \((3^(-1))^(2x)=3^((-1) 2x) =3^(-2x)\).

\(3^(x+0.5)=3^(-2x)\)

And now our bases are equal and there are no interfering coefficients, etc. So we can make the transition.

Example . Solve the exponential equation \(4^(x+0.5)-5 2^x+2=0\)
Solution:

\(4^(x+0.5)-5 2^x+2=0\)		We again use the power property \(a^b \cdot a^c=a^(b+c)\) in the opposite direction.
\(4^x 4^(0.5)-5 2^x+2=0\)		Now remember that \(4=2^2\).
\((2^2)^x·(2^2)^(0.5)-5·2^x+2=0\)		Using the properties of degrees, we transform: \((2^2)^x=2^(2x)=2^(x 2)=(2^x)^2\) \((2^2)^(0.5)=2^(2 0.5)=2^1=2.\)
\(2·(2^x)^2-5·2^x+2=0\)		We look carefully at the equation and see that the replacement \(t=2^x\) suggests itself.
\(t_1=2\) \(t_2=\frac(1)(2)\)		However, we have found the values ​​of \(t\), and we need \(x\). We return to the X's, making a reverse replacement.
\(2^x=2\) \(2^x=\frac(1)(2)\)		Let's transform the second equation using the negative power property...
\(2^x=2^1\) \(2^x=2^(-1)\)		...and we decide until the answer.
\(x_1=1\) \(x_2=-1\)

Answer : \(-1; 1\).

The question remains - how to understand when to use which method? This comes with experience. Until you have developed it, use the general recommendation for solving complex problems - “if you don’t know what to do, do what you can.” That is, look for how you can transform the equation in principle, and try to do it - what if what happens? The main thing is to make only mathematically based transformations.

Exponential equations without solutions

Let's look at two more situations that often confuse students:
- a positive number to the power is equal to zero, for example, \(2^x=0\);
- a positive number is equal to a power of a negative number, for example, \(2^x=-4\).

Let's try to solve by brute force. If x is a positive number, then as x grows, the entire power \(2^x\) will only increase:

\(x=1\); \(2^1=2\)
\(x=2\); \(2^2=4\)
\(x=3\); \(2^3=8\).

\(x=0\); \(2^0=1\)

Also by. Negative X's remain. Remembering the property \(a^(-n)=\frac(1)(a^n)\), we check:

\(x=-1\); \(2^(-1)=\frac(1)(2^1) =\frac(1)(2)\)
\(x=-2\); \(2^(-2)=\frac(1)(2^2) =\frac(1)(4)\)
\(x=-3\); \(2^(-3)=\frac(1)(2^3) =\frac(1)(8)\)

Despite the fact that the number becomes smaller with each step, it will never reach zero. So the negative degree did not save us. We come to a logical conclusion:

A positive number to any degree will remain a positive number.

Thus, both equations above have no solutions.

Exponential equations with different bases

In practice, sometimes we encounter exponential equations with different bases that are not reducible to each other, and at the same time with the same exponents. They look like this: \(a^(f(x))=b^(f(x))\), where \(a\) and \(b\) are positive numbers.

For example:

\(7^(x)=11^(x)\)
\(5^(x+2)=3^(x+2)\)
\(15^(2x-1)=(\frac(1)(7))^(2x-1)\)

Such equations can easily be solved by dividing by any of the sides of the equation (usually divided by the right side, that is, by \(b^(f(x))\). You can divide this way because a positive number is positive to any power (that is, we do not divide by zero) We get:

\(\frac(a^(f(x)))(b^(f(x)))\) \(=1\)

Example . Solve the exponential equation \(5^(x+7)=3^(x+7)\)
Solution:

\(5^(x+7)=3^(x+7)\)		Here we won’t be able to turn a five into a three, or vice versa (at least without using ). This means we cannot come to the form \(a^(f(x))=a^(g(x))\). However, the indicators are the same. Let's divide the equation by the right side, that is, by \(3^(x+7)\) (we can do this because we know that three will not be zero to any degree).
\(\frac(5^(x+7))(3^(x+7))\) \(=\)\(\frac(3^(x+7))(3^(x+7) )\)		Now remember the property \((\frac(a)(b))^c=\frac(a^c)(b^c)\) and use it from the left in the opposite direction. On the right, we simply reduce the fraction.
\((\frac(5)(3))^(x+7)\) \(=1\)		It would seem that things didn't get any better. But remember one more property of power: \(a^0=1\), in other words: “any number to the zero power is equal to \(1\).” The converse is also true: “one can be represented as any number to the zero power.” Let's take advantage of this by making the base on the right the same as on the left.
\((\frac(5)(3))^(x+7)\) \(=\) \((\frac(5)(3))^0\)		Voila! Let's get rid of the bases.
		We are writing a response.

Answer : \(-7\).

Sometimes the “sameness” of exponents is not obvious, but skillful use of the properties of exponents resolves this issue.

Example . Solve the exponential equation \(7^( 2x-4)=(\frac(1)(3))^(-x+2)\)
Solution:

\(7^( 2x-4)=(\frac(1)(3))^(-x+2)\)		The equation looks very sad... Not only can the bases not be reduced to the same number (seven will in no way be equal to \(\frac(1)(3)\)), but also the exponents are different... However, let's use the left exponent deuce.
\(7^( 2(x-2))=(\frac(1)(3))^(-x+2)\)		Remembering the property \((a^b)^c=a^(b·c)\) , we transform from the left: \(7^(2(x-2))=7^(2·(x-2))=(7^2)^(x-2)=49^(x-2)\).
\(49^(x-2)=(\frac(1)(3))^(-x+2)\)		Now, remembering the property of negative degree \(a^(-n)=\frac(1)(a)^n\), we transform from the right: \((\frac(1)(3))^(-x+2) =(3^(-1))^(-x+2)=3^(-1(-x+2))=3^(x-2)\)
\(49^(x-2)=3^(x-2)\)		Hallelujah! The indicators are the same! Acting according to the scheme already familiar to us, we solve before the answer.

Answer : \(2\).