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How to solve equations with an unknown to a power. Solving exponential equations

At the stage of preparation for the final test, high school students need to improve their knowledge on the topic “Exponential Equations.” The experience of past years indicates that such tasks cause certain difficulties for schoolchildren. Therefore, high school students, regardless of their level of preparation, need to thoroughly master the theory, remember the formulas and understand the principle of solving such equations. Having learned to cope with this type of problem, graduates can count on high scores when passing the Unified State Exam in mathematics.

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

The Shkolkovo educational portal invites students to use our knowledge base. We implement completely new method preparation for the final test. By studying on our website, you will be able to identify gaps in knowledge and pay attention to those tasks that cause the most difficulty.

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

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

To better understand the material, we recommend that you practice completing the assignments. Carefully review the examples presented on this page. exponential equations with the solution to understand the calculation algorithm. After that, proceed to perform tasks in the “Directories” section. You can start with the easiest tasks or go straight to solving complex exponential equations with several unknowns or . The database of exercises on our website is constantly supplemented and updated.

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In this lesson we will look at solving more complex exponential equations, recall the basic theoretical principles regarding exponential function.

1. Definition and properties of the exponential function, methods for solving the simplest exponential equations

Let us recall the definition and basic properties of the exponential function. The solution of all exponential equations and inequalities is based on these properties.

Exponential function is a function of the form , where the base is the degree and Here x is the independent variable, argument; y is the dependent variable, function.

Rice. 1. Graph of exponential function

The graph shows increasing and decreasing exponents, illustrating the exponential function with a base greater than one and less than one but greater than zero, respectively.

Both curves pass through the point (0;1)

Properties of the Exponential Function:

Domain: ;

Range of values: ;

The function is monotonic, increases with, decreases with.

A monotonic function takes each of its values given a single argument value.

When the argument increases from minus to plus infinity, the function increases from zero inclusive to plus infinity. On the contrary, when the argument increases from minus to plus infinity, the function decreases from infinity to zero, not inclusive.

2. Solving standard exponential equations

Let us remind you how to solve the simplest exponential equations. Their solution is based on the monotonicity of the exponential function. Almost all complex exponential equations can be reduced to such equations.

The equality of exponents with equal bases is due to the property of the exponential function, namely its monotonicity.

Solution method:

Equalize the bases of degrees;

Equate the exponents.

Let's move on to consider more complex exponential equations; our goal is to reduce each of them to the simplest.

Let's get rid of the root on the left side and bring the degrees to the same base:

In order to reduce a complex exponential equation to its simplest, substitution of variables is often used.

Let's use the power property:

We are introducing a replacement. Let it be then

Let's multiply the resulting equation by two and move all terms to the left side:

The first root does not satisfy the range of y values, so we discard it. We get:

Let's reduce the degrees to the same indicator:

Let's introduce a replacement:

Let it be then . With such a replacement, it is obvious that y accepts strictly positive values. We get:

We know how to solve such quadratic equations, we can write down the answer:

To make sure that the roots are found correctly, you can check using Vieta’s theorem, i.e., find the sum of the roots and their product and compare them with the corresponding coefficients of the equation.

We get:

3. Methodology for solving homogeneous exponential equations of the second degree

Let's study the following important type of exponential equations:

Equations of this type are called homogeneous of the second degree with respect to the functions f and g. On its left side there is a square trinomial with respect to f with the parameter g or a square trinomial with respect to g with the parameter f.

Solution method:

This equation can be solved as a quadratic equation, but it is easier to do it differently. There are two cases to consider:

In the first case we get

In the second case, we have the right to divide by the highest degree and get:

We should introduce a change of variables, we get quadratic equation relative to y:

Let us note that the functions f and g can be any, but we are interested in the case when these are exponential functions.

4. Examples of solving homogeneous equations

Let's move all the terms to the left side of the equation:

Since exponential functions acquire strictly positive values, we have the right to immediately divide the equation by , without considering the case when:

We get:

Let's introduce a replacement: (according to the properties of the exponential function)

We got a quadratic equation:

We determine the roots using Vieta’s theorem:

The first root does not satisfy the range of values of y, we discard it, we get:

Let's use the properties of degrees and reduce all degrees to simple bases:

It's easy to notice the functions f and g:

Since exponential functions acquire strictly positive values, we have the right to immediately divide the equation by , without considering the case when .

Solving exponential equations. Examples.

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

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

There you are examples of exponential equations:

3 x 2 x = 8 x+3

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

this will 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 actually just threw it away identical grounds(threes). 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 common grounds under different numbers) is a very popular technique in exponential equations! Yes, and in logarithms too. You must be able to recognize powers of other numbers in numbers. This is extremely important for solving exponential equations.

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

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

Determine what powers and what numbers the numbers are:

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

Answers (in a mess, of course!):

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

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

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

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

3 2x+4 -11 9 x = 210

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

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

Using the same rules for dealing with degrees:

3 2x+4 = 3 2x ·3 4

That’s great, you can write it down:

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

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

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

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

Look, everything will work out).

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

3 2x (3 4 - 11) = 210

3 4 - 11 = 81 - 11 = 70

The example keeps getting better and better!

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

Oops! Everything got better!

This is the final answer.

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

Replacing a variable in solving exponential equations. Examples.

Let's solve the equation:

4 x - 3 2 x +2 = 0

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

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

We get the equation:

2 2x - 3 2 x +2 = 0

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

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

So let

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

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

Well, 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:

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

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

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

Practical advice:

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

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

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

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

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

Solve exponential equations:

More difficult:

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

9 x - 8 3 x = 9

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

Find the product of roots:

2 3's + 2 x = 9

Happened?

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

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

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

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

A simpler example, for relaxation):

9 2 x - 4 3 x = 0

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

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

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

Answers (in disarray, separated by semicolons):

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

Is everything successful? Great.

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

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

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.

Exponential equations are those in which the unknown is contained in the exponent. The simplest exponential equation has the form: a x = a b, where a> 0, a 1, x is unknown.

The main properties of powers by which exponential equations are transformed: a>0, b>0.

When solving exponential equations, the following properties of the exponential function are also used: y = a x, a > 0, a1:

To represent a number as a power, use the basic logarithmic identity: b = , a > 0, a1, b > 0.

Problems and tests on the topic "Exponential Equations"

Exponential equations
Lessons: 4 Assignments: 21 Tests: 1
Exponential equations - Important topics for reviewing the Unified State Examination in mathematics
Tasks: 14
Systems of exponential and logarithmic equations - Demonstrative and logarithmic functions Grade 11
Lessons: 1 Assignments: 15 Tests: 1
§2.1. Solving exponential equations
Lessons: 1 Tasks: 27
§7 Exponential and logarithmic equations and inequalities - Section 5. Exponential and logarithmic functions, grade 10
Lessons: 1 Tasks: 17

To successfully solve exponential equations, you must know the basic properties of powers, properties of the exponential function, and the basic logarithmic identity.

When solving exponential equations, two main methods are used:

transition from the equation a f(x) = a g(x) to the equation f(x) = g(x);
introduction of new lines.

Examples.

1. Equations reduced to the simplest. They are solved by reducing both sides of the equation to a power with the same base.

3 x = 9 x – 2 .

Solution:

3 x = (3 2) x – 2 ;
3 x = 3 2x – 4 ;
x = 2x –4;
x = 4.

Answer: 4.

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

Solution:

3 x – 3 x – 2 = 24
3 x – 2 (3 2 – 1) = 24
3 x – 2 × 8 = 24
3 x – 2 = 3
x – 2 = 1
x = 3.

Answer: 3.

3. Equations solved using a change of variable.

Solution:

2 2x + 2 x – 12 = 0
We denote 2 x = y.
y 2 + y – 12 = 0
y 1 = - 4; y2 = 3.
a) 2 x = - 4. The equation has no solutions, because 2 x > 0.
b) 2 x = 3; 2 x = 2 log 2 3 ; x = log 2 3.

Answer: log 2 3.

4. Equations containing powers with two different (not reducible to each other) bases.

3 × 2 x + 1 - 2 × 5 x – 2 = 5 x + 2 x – 2.

3× 2 x + 1 – 2 x – 2 = 5 x – 2 × 5 x – 2
2 x – 2 ×23 = 5 x – 2
×23
2 x – 2 = 5 x – 2
(5/2) x– 2 = 1
x – 2 = 0
x = 2.

Answer: 2.

5. Equations that are homogeneous with respect to a x and b x.

General form: .

9 x + 4 x = 2.5 × 6 x.

Solution:

3 2x – 2.5 × 2 x × 3 x +2 2x = 0 |: 2 2x > 0
(3/2) 2x – 2.5 × (3/2) x + 1 = 0.
Let us denote (3/2) x = y.
y 2 – 2.5y + 1 = 0,
y 1 = 2; y 2 = ½.

Answer: log 3/2 2; - log 3/2 2.

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

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

Before you start solving an exponential equation, it is useful to do a few things preliminary actions , which can significantly facilitate the process of solving it. These are the steps:

1. Divide all bases of powers into prime factors.

2. Present the roots as a degree.

3. Decimals imagine as ordinary ones.

4. Mixed numbers write as improper fractions.

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

Let's look at the main types exponential equations and algorithms for solving them.

1. Equation of the form

This equation is equivalent to the equation

Watch the solution to the equation in this VIDEO TUTORIAL this type.

2. Equation of the form

In equations of this type:

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

To solve this equation, you need to factor out the smallest factor.

An example of solving an equation of this type:

watch the VIDEO TUTORIAL.

3. Equation of the form

Equations of this type differ in that

a) all degrees have the same bases

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

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

Watch the VIDEO LESSON to solve this type of equation:

4. Homogeneous equations kind

Distinctive features of homogeneous equations:

a) all monomials have the same degree,

b) the free term is zero,

c) the equation contains powers with two different bases.

Homogeneous equations are solved using a similar algorithm.

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

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

In our case, since the expression is not zero for any value of the unknown, we can divide by it without fear. Let's divide the left side of the equation by this expression term by term. We get: