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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 identical 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 powers.
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 it is clear that on the left and right sides the bases are the same and equal to three, which means we can discard them and equate the degrees.
3x=2x+16 we get the simplest equation
3x - 2x=16
x=16
Answer: x=16.
Let's look at the following example:
2 2x+4 - 10 4 x = 2 4
First of all, we look at the bases, bases two and four. And we need them to be the same. We transform the four using the formula (a n) m = a nm.
4 x = (2 2) x = 2 2x
And we also use one formula a n a m = a n + m:
2 2x+4 = 2 2x 2 4
Add to the equation:
2 2x 2 4 - 10 2 2x = 24
We gave an example for the same reasons. But other numbers 10 and 24 bother us. What to do with them? If you look closely you can see that on the left side we have 2 2x repeated, here is the answer - we can put 2 2x out of brackets:
2 2x (2 4 - 10) = 24
Let's calculate the expression in brackets:
2 4 — 10 = 16 — 10 = 6
We divide the entire equation by 6:
Let's imagine 4=2 2:
2 2x = 2 2 bases are the same, we discard them and equate the degrees.
2x = 2 is the simplest equation. Divide it by 2 and we get
x = 1
Answer: x = 1.
Let's solve the equation:
9 x – 12*3 x +27= 0
Let's transform:
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
Therefore,
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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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! This is important.
Here you go examples of exponential equations:
3 x 2 x = 8 x+3
Pay attention! 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 identical 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 give it. But now you know where to aim when solving tricky examples. It must be brought to the form where the same base number is on the left and right. Then everything will be easier. Actually, this is a classic of mathematics. We take the original example and transform it to the desired one us mind. According to the rules of mathematics, of course.
Let's look at examples that require some additional effort to reduce them to the simplest. Let's call them simple exponential equations.
Solving simple exponential equations. Examples.
When solving exponential equations, the main rules are actions with degrees. Without knowledge of these actions nothing will work.
To actions with degrees, one must add personal observation and ingenuity. Do we need the same base numbers? So we look for them in the example in explicit or encrypted form.
Let's see how this is done in practice?
Let us be given an example:
2 2x - 8 x+1 = 0
The first keen glance is at grounds. They... They are different! Two and eight. But it’s too early to become discouraged. It's time to remember that
Two and eight are relatives in degree.) It is quite possible to write:
8 x+1 = (2 3) x+1
If we recall the formula from operations with degrees:
(a n) m = a nm ,
this works out great:
8 x+1 = (2 3) x+1 = 2 3(x+1)
Original example started 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 (encoding common bases under different numbers) is a very popular technique in exponential equations! Yes, and in logarithms too. You need to 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 to high school straight away, 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. And 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. And what 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 get another powerful and universal method. 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:
Therefore,
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 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.
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 identical numbers in any powers. We use actions with degrees And factorization. What can be counted in numbers, we count.
3. If the second tip did not 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
Did it work?
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 worthy of increased difficulty. Let me hint that in this example, ingenuity and the most universal rule solutions to 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.
Any problems? No question! 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.
This is the name for equations of the form where the unknown is in both the exponent and the base of the power.
You can specify a completely clear algorithm for solving an equation of the form. To do this, you need to pay attention to the fact that when Oh) Not equal to zero, one and minus one, equality of degrees with the same bases (be it positive or negative) is possible only if the exponents are equal. That is, all the roots of the equation will be the roots of the equation f(x) = g(x) The converse statement is not true, when Oh)< 0 and fractional values f(x) And g(x) expressions Oh) f(x) And
Oh) g(x) lose their meaning. That is, when moving from to f(x) = g(x)(for and extraneous roots may appear, which must be excluded by checking against the original equation. And cases a = 0, a = 1, a = -1 need to be considered separately.
So, to completely solve the equation, we consider the cases:
a(x) = O f(x) And g(x) will be positive numbers, then this is the solution. Otherwise, no
a(x) = 1. The roots of this equation are also the roots of the original equation.
a(x) = -1. If, for a value of x that satisfies this equation, f(x) And g(x) are integers of the same parity (either both even or both odd), then this is the solution. Otherwise, no
When and we solve the equation f(x)= g(x) and by substituting the obtained results into the original equation we cut off the extraneous roots.
Examples of solving exponential-power equations.
Example No. 1.
1) x - 3 = 0, x = 3. because 3 > 0, and 3 2 > 0, then x 1 = 3 is the solution.
2) x - 3 = 1, x 2 = 4.
3) x - 3 = -1, x = 2. Both indicators are even. This solution is x 3 = 1.
4) x - 3 ? 0 and x? ± 1. x = x 2, x = 0 or x = 1. For x = 0, (-3) 0 = (-3) 0 - this solution is correct: x 4 = 0. For x = 1, (-2) 1 = (-2) 1 - this solution is correct x 5 = 1.
Answer: 0, 1, 2, 3, 4.
Example No. 2.
By definition of an arithmetic square root: x - 1? 0, x? 1.
1) x - 1 = 0 or x = 1, = 0, 0 0 is not a solution.
2) x - 1 = 1 x 1 = 2.
3) x - 1 = -1 x 2 = 0 does not fit in ODZ.
D = (-2) - 4*1*5 = 4 - 20 = -16 - there are no roots.
This lesson is intended for those who are just beginning to learn exponential equations. As always, let's start with the definition and simple examples.
If you are reading this lesson, then I suspect that you already have at least a minimal understanding of the simplest equations - linear and quadratic: $56x-11=0$; $((x)^(2))+5x+4=0$; $((x)^(2))-12x+32=0$, etc. Being able to solve such constructions is absolutely necessary in order not to “get stuck” in the topic that will now be discussed.
So, exponential equations. Let me give you a couple of examples:
\[((2)^(x))=4;\quad ((5)^(2x-3))=\frac(1)(25);\quad ((9)^(x))=- 3\]
Some of them may seem more complex to you, while others, on the contrary, are too simple. But they all have one important feature in common: their notation contains the exponential function $f\left(x \right)=((a)^(x))$. Thus, let's introduce the definition:
An exponential equation is any equation containing an exponential function, i.e. expression of the form $((a)^(x))$. In addition to the specified function, such equations can contain any other algebraic constructions - polynomials, roots, trigonometry, logarithms, etc.
OK then. We've sorted out the definition. Now the question is: how to solve all this crap? The answer is both simple and complex.
Let's start with the good news: from my experience of teaching many students, I can say that most of them find exponential equations much easier than the same logarithms, and even more so trigonometry.
But there is also bad news: sometimes the compilers of problems for all kinds of textbooks and exams are struck by “inspiration”, and their drug-inflamed brain begins to produce such brutal equations that solving them becomes problematic not only for students - even many teachers get stuck on such problems.
However, let's not talk about sad things. And let's return to those three equations that were given at the very beginning of the story. Let's try to solve each of them.
First equation: $((2)^(x))=4$. Well, to what power do you need to raise the number 2 to get the number 4? Probably the second? After all, $((2)^(2))=2\cdot 2=4$ - and we got the correct numerical equality, i.e. indeed $x=2$. Well, thanks, Cap, but this equation was so simple that even my cat could solve it. :)
Let's look at the following equation:
\[((5)^(2x-3))=\frac(1)(25)\]
But here it’s a little more complicated. Many students know that $((5)^(2))=25$ is the multiplication table. Some also suspect that $((5)^(-1))=\frac(1)(5)$ is essentially the definition of negative powers (similar to the formula $((a)^(-n))= \frac(1)(((a)^(n)))$).
Finally, only a select few realize that these facts can be combined and yield the following result:
\[\frac(1)(25)=\frac(1)(((5)^(2)))=((5)^(-2))\]
Thus, our original equation will be rewritten as follows:
\[((5)^(2x-3))=\frac(1)(25)\Rightarrow ((5)^(2x-3))=((5)^(-2))\]
But this is already completely solvable! On the left in the equation there is an exponential function, on the right in the equation there is an exponential function, there is nothing else anywhere except them. Therefore, we can “discard” the bases and stupidly equate the indicators:
We have obtained the simplest linear equation that any student can solve in just a couple of lines. Okay, in four lines:
\[\begin(align)& 2x-3=-2 \\& 2x=3-2 \\& 2x=1 \\& x=\frac(1)(2) \\\end(align)\]
If you don’t understand what was happening in the last four lines, be sure to return to the topic “ linear equations"and repeat it. Because without a clear understanding of this topic, it is too early for you to take on exponential equations.
\[((9)^(x))=-3\]
So how can we solve this? First thought: $9=3\cdot 3=((3)^(2))$, so the original equation can be rewritten as follows:
\[((\left(((3)^(2)) \right))^(x))=-3\]
Then we remember that when raising a power to a power, the exponents are multiplied:
\[((\left(((3)^(2)) \right))^(x))=((3)^(2x))\Rightarrow ((3)^(2x))=-(( 3)^(1))\]
\[\begin(align)& 2x=-1 \\& x=-\frac(1)(2) \\\end(align)\]
And for such a decision we will receive a honestly deserved two. For, with the equanimity of a Pokemon, we sent the minus sign in front of the three to the power of this very three. But you can’t do that. And here's why. Take a look at the different powers of three:
\[\begin(matrix) ((3)^(1))=3& ((3)^(-1))=\frac(1)(3)& ((3)^(\frac(1)( 2)))=\sqrt(3) \\ ((3)^(2))=9& ((3)^(-2))=\frac(1)(9)& ((3)^(\ frac(1)(3)))=\sqrt(3) \\ ((3)^(3))=27& ((3)^(-3))=\frac(1)(27)& (( 3)^(-\frac(1)(2)))=\frac(1)(\sqrt(3)) \\\end(matrix)\]
When compiling this tablet, I did not pervert anything: I considered positive powers, and negative ones, and even fractional ones... well, where is at least one negative number here? He's gone! And it cannot be, because the exponential function $y=((a)^(x))$, firstly, always takes only positive values(no matter how much you multiply one or divide it by two, it will still be a positive number), and secondly, the base of such a function - the number $a$ - is by definition a positive number!
Well, then how to solve the equation $((9)^(x))=-3$? But no way: there are no roots. And in this sense, exponential equations are very similar to quadratic equations - there may also be no roots. But if in quadratic equations the number of roots is determined by the discriminant (positive discriminant - 2 roots, negative - no roots), then in exponential equations everything depends on what is to the right of the equal sign.
Thus, let us formulate the key conclusion: the simplest exponential equation of the form $((a)^(x))=b$ has a root if and only if $b>0$. Knowing this simple fact, you can easily determine whether the equation proposed to you has roots or not. Those. Is it worth solving it at all or immediately writing down that there are no roots.
This knowledge will help us many times when we have to solve more complex problems. For now, enough of the lyrics - it’s time to study the basic algorithm for solving exponential equations.
How to Solve Exponential Equations
So, let's formulate the problem. It is necessary to solve the exponential equation:
\[((a)^(x))=b,\quad a,b>0\]
According to the “naive” algorithm that we used earlier, it is necessary to represent the number $b$ as a power of the number $a$:
In addition, if instead of the variable $x$ there is any expression, we will get a new equation that can already be solved. For example:
\[\begin(align)& ((2)^(x))=8\Rightarrow ((2)^(x))=((2)^(3))\Rightarrow x=3; \\& ((3)^(-x))=81\Rightarrow ((3)^(-x))=((3)^(4))\Rightarrow -x=4\Rightarrow x=-4; \\& ((5)^(2x))=125\Rightarrow ((5)^(2x))=((5)^(3))\Rightarrow 2x=3\Rightarrow x=\frac(3)( 2). \\\end(align)\]
And oddly enough, this scheme works in about 90% of cases. What then about the remaining 10%? The remaining 10% are slightly “schizophrenic” exponential equations of the form:
\[((2)^(x))=3;\quad ((5)^(x))=15;\quad ((4)^(2x))=11\]
Well, to what power do you need to raise 2 to get 3? First? But no: $((2)^(1))=2$ is not enough. Second? No either: $((2)^(2))=4$ is too much. Which one then?
Knowledgeable students have probably already guessed: in such cases, when it is not possible to solve “beautifully”, the “heavy artillery” - logarithms - comes into play. Let me remind you that using logarithms, any positive number can be represented as a power of any other positive number (except for one):
Remember this formula? When I tell my students about logarithms, I always warn: this formula (it is also the main logarithmic identity or, if you like, the definition of a logarithm) will haunt you for a very long time and “pop up” in the most unexpected places. Well, she surfaced. Let's look at our equation and this formula:
\[\begin(align)& ((2)^(x))=3 \\& a=((b)^(((\log )_(b))a)) \\\end(align) \]
If we assume that $a=3$ is our original number on the right, and $b=2$ is the very base of the exponential function to which we so want to reduce the right-hand side, we get the following:
\[\begin(align)& a=((b)^(((\log )_(b))a))\Rightarrow 3=((2)^(((\log )_(2))3 )); \\& ((2)^(x))=3\Rightarrow ((2)^(x))=((2)^(((\log )_(2))3))\Rightarrow x=( (\log )_(2))3. \\\end(align)\]
We received a slightly strange answer: $x=((\log )_(2))3$. In some other task, many would have doubts with such an answer and would begin to double-check their solution: what if an error had crept in somewhere? I hasten to please you: there is no error here, and logarithms in the roots of exponential equations are a completely typical situation. So get used to it. :)
Now let’s solve the remaining two equations by analogy:
\[\begin(align)& ((5)^(x))=15\Rightarrow ((5)^(x))=((5)^(((\log )_(5))15)) \Rightarrow x=((\log )_(5))15; \\& ((4)^(2x))=11\Rightarrow ((4)^(2x))=((4)^(((\log )_(4))11))\Rightarrow 2x=( (\log )_(4))11\Rightarrow x=\frac(1)(2)((\log )_(4))11. \\\end(align)\]
That's it! By the way, the last answer can be written differently:
We introduced a factor into the argument of the logarithm. But no one is stopping us from adding this factor to the base:
Moreover, all three options are correct - it’s simple different shapes records of the same number. Which one to choose and write down in this solution is up to you to decide.
Thus, we have learned to solve any exponential equations of the form $((a)^(x))=b$, where the numbers $a$ and $b$ are strictly positive. However, the harsh reality of our world is that such simple tasks you will meet very, very rarely. More often than not you will come across something like this:
\[\begin(align)& ((4)^(x))+((4)^(x-1))=((4)^(x+1))-11; \\& ((7)^(x+6))\cdot ((3)^(x+6))=((21)^(3x)); \\& ((100)^(x-1))\cdot ((2.7)^(1-x))=0.09. \\\end(align)\]
So how can we solve this? Can this be solved at all? And if so, how?
Don't panic. All these equations quickly and easily reduce to the simple formulas that we have already considered. You just need to remember a couple of tricks from the algebra course. And of course, there are no rules for working with degrees. I'll tell you about all this now. :)
Converting Exponential Equations
The first thing to remember: any exponential equation, no matter how complex it may be, one way or another must be reduced to the simplest equations - the ones that we have already considered and which we know how to solve. In other words, the solution scheme for any exponential equation looks like this:
- Write down the original equation. For example: $((4)^(x))+((4)^(x-1))=((4)^(x+1))-11$;
- Do some weird shit. Or even some crap called "convert an equation";
- At the output, get the simplest expressions of the form $((4)^(x))=4$ or something else like that. Moreover, one initial equation can give several such expressions at once.
Everything is clear with the first point - even my cat can write the equation on a piece of paper. The third point also seems to be more or less clear - we have already solved a whole bunch of such equations above.
But what about the second point? What kind of transformations? Convert what into what? And how?
Well, let's find out. First of all, I would like to note the following. All exponential equations are divided into two types:
- The equation is composed of exponential functions with the same base. Example: $((4)^(x))+((4)^(x-1))=((4)^(x+1))-11$;
- The formula contains exponential functions with different bases. Examples: $((7)^(x+6))\cdot ((3)^(x+6))=((21)^(3x))$ and $((100)^(x-1) )\cdot ((2,7)^(1-x))=$0.09.
Let's start with equations of the first type - they are the easiest to solve. And in solving them, we will be helped by such a technique as highlighting stable expressions.
Isolating a stable expression
Let's look at this equation again:
\[((4)^(x))+((4)^(x-1))=((4)^(x+1))-11\]
What do we see? The four are raised to different degrees. But all these powers are simple sums of the variable $x$ with other numbers. Therefore, it is necessary to remember the rules for working with degrees:
\[\begin(align)& ((a)^(x+y))=((a)^(x))\cdot ((a)^(y)); \\& ((a)^(x-y))=((a)^(x)):((a)^(y))=\frac(((a)^(x)))(((a )^(y))). \\\end(align)\]
Simply put, addition can be converted to a product of powers, and subtraction can easily be converted to division. Let's try to apply these formulas to the degrees from our equation:
\[\begin(align)& ((4)^(x-1))=\frac(((4)^(x)))(((4)^(1)))=((4)^ (x))\cdot \frac(1)(4); \\& ((4)^(x+1))=((4)^(x))\cdot ((4)^(1))=((4)^(x))\cdot 4. \ \\end(align)\]
Let's rewrite the original equation taking this fact into account, and then collect all the terms on the left:
\[\begin(align)& ((4)^(x))+((4)^(x))\cdot \frac(1)(4)=((4)^(x))\cdot 4 -11; \\& ((4)^(x))+((4)^(x))\cdot \frac(1)(4)-((4)^(x))\cdot 4+11=0. \\\end(align)\]
The first four terms contain the element $((4)^(x))$ - let’s take it out of the bracket:
\[\begin(align)& ((4)^(x))\cdot \left(1+\frac(1)(4)-4 \right)+11=0; \\& ((4)^(x))\cdot \frac(4+1-16)(4)+11=0; \\& ((4)^(x))\cdot \left(-\frac(11)(4) \right)=-11. \\\end(align)\]
It remains to divide both sides of the equation by the fraction $-\frac(11)(4)$, i.e. essentially multiply by the inverted fraction - $-\frac(4)(11)$. We get:
\[\begin(align)& ((4)^(x))\cdot \left(-\frac(11)(4) \right)\cdot \left(-\frac(4)(11) \right )=-11\cdot \left(-\frac(4)(11) \right); \\& ((4)^(x))=4; \\& ((4)^(x))=((4)^(1)); \\& x=1. \\\end(align)\]
That's it! We have reduced the original equation to its simplest form and obtained the final answer.
At the same time, in the process of solving we discovered (and even took it out of the bracket) the common factor $((4)^(x))$ - this is a stable expression. It can be designated as a new variable, or you can simply express it carefully and get the answer. Anyway, key principle The solutions are as follows:
Find in the original equation a stable expression containing a variable that is easily distinguished from all exponential functions.
The good news is that almost every exponential equation allows you to isolate such a stable expression.
But the bad news is that these expressions can be quite tricky and can be quite difficult to identify. So let's look at one more problem:
\[((5)^(x+2))+((0,2)^(-x-1))+4\cdot ((5)^(x+1))=2\]
Perhaps someone will now have a question: “Pasha, are you stoned? There are different bases here – 5 and 0.2.” But let's try converting the power to base 0.2. For example, let’s get rid of the decimal fraction by reducing it to a regular one:
\[((0,2)^(-x-1))=((0,2)^(-\left(x+1 \right)))=((\left(\frac(2)(10 ) \right))^(-\left(x+1 \right)))=((\left(\frac(1)(5) \right))^(-\left(x+1 \right)) )\]
As you can see, the number 5 still appeared, albeit in the denominator. At the same time, the indicator was rewritten as negative. And now let's remember one of the most important rules work with degrees:
\[((a)^(-n))=\frac(1)(((a)^(n)))\Rightarrow ((\left(\frac(1)(5) \right))^( -\left(x+1 \right)))=((\left(\frac(5)(1) \right))^(x+1))=((5)^(x+1))\ ]
Here, of course, I was lying a little. Because for a complete understanding, the formula for getting rid of negative indicators had to be written like this:
\[((a)^(-n))=\frac(1)(((a)^(n)))=((\left(\frac(1)(a) \right))^(n ))\Rightarrow ((\left(\frac(1)(5) \right))^(-\left(x+1 \right)))=((\left(\frac(5)(1) \ right))^(x+1))=((5)^(x+1))\]
On the other hand, nothing prevented us from working with just fractions:
\[((\left(\frac(1)(5) \right))^(-\left(x+1 \right)))=((\left(((5)^(-1)) \ right))^(-\left(x+1 \right)))=((5)^(\left(-1 \right)\cdot \left(-\left(x+1 \right) \right) ))=((5)^(x+1))\]
But in this case, you need to be able to raise a power to another power (let me remind you: in this case, the indicators are added together). But I didn’t have to “reverse” the fractions - perhaps this will be easier for some. :)
In any case, the original exponential equation will be rewritten as:
\[\begin(align)& ((5)^(x+2))+((5)^(x+1))+4\cdot ((5)^(x+1))=2; \\& ((5)^(x+2))+5\cdot ((5)^(x+1))=2; \\& ((5)^(x+2))+((5)^(1))\cdot ((5)^(x+1))=2; \\& ((5)^(x+2))+((5)^(x+2))=2; \\& 2\cdot ((5)^(x+2))=2; \\& ((5)^(x+2))=1. \\\end(align)\]
So it turns out that the original equation can be solved even more simply than the one previously considered: here you don’t even need to select a stable expression - everything has been reduced by itself. It remains only to remember that $1=((5)^(0))$, from which we get:
\[\begin(align)& ((5)^(x+2))=((5)^(0)); \\& x+2=0; \\& x=-2. \\\end(align)\]
That's the solution! We got the final answer: $x=-2$. At the same time, I would like to note one technique that greatly simplified all calculations for us:
In exponential equations, be sure to get rid of decimal fractions and convert them to ordinary ones. This will allow you to see the same bases of degrees and greatly simplify the solution.
Let us now move on to more complex equations in which there are different bases that cannot be reduced to each other using powers at all.
Using the Degrees Property
Let me remind you that we have two more particularly harsh equations:
\[\begin(align)& ((7)^(x+6))\cdot ((3)^(x+6))=((21)^(3x)); \\& ((100)^(x-1))\cdot ((2.7)^(1-x))=0.09. \\\end(align)\]
The main difficulty here is that it is not clear what to give and to what basis. Where set expressions? Where are the same grounds? There is none of this.
But let's try to go a different way. If there are no ready-made identical bases, you can try to find them by factoring the existing bases.
Let's start with the first equation:
\[\begin(align)& ((7)^(x+6))\cdot ((3)^(x+6))=((21)^(3x)); \\& 21=7\cdot 3\Rightarrow ((21)^(3x))=((\left(7\cdot 3 \right))^(3x))=((7)^(3x))\ cdot((3)^(3x)). \\\end(align)\]
But you can do the opposite - make the number 21 from the numbers 7 and 3. This is especially easy to do on the left, since the indicators of both degrees are the same:
\[\begin(align)& ((7)^(x+6))\cdot ((3)^(x+6))=((\left(7\cdot 3 \right))^(x+ 6))=((21)^(x+6)); \\& ((21)^(x+6))=((21)^(3x)); \\& x+6=3x; \\& 2x=6; \\& x=3. \\\end(align)\]
That's it! You took the exponent outside the product and immediately got a beautiful equation that can be solved in a couple of lines.
Now let's look at the second equation. Everything is much more complicated here:
\[((100)^(x-1))\cdot ((2.7)^(1-x))=0.09\]
\[((100)^(x-1))\cdot ((\left(\frac(27)(10) \right))^(1-x))=\frac(9)(100)\]
IN in this case the fractions turned out to be irreducible, but if something could be reduced, be sure to reduce it. Often, interesting reasons will appear with which you can already work.
Unfortunately, nothing special appeared for us. But we see that the exponents on the left in the product are opposite:
Let me remind you: to get rid of the minus sign in the indicator, you just need to “flip” the fraction. Well, let's rewrite the original equation:
\[\begin(align)& ((100)^(x-1))\cdot ((\left(\frac(10)(27) \right))^(x-1))=\frac(9 )(100); \\& ((\left(100\cdot \frac(10)(27) \right))^(x-1))=\frac(9)(100); \\& ((\left(\frac(1000)(27) \right))^(x-1))=\frac(9)(100). \\\end(align)\]
In the second line we simply carried out general indicator from the product out of brackets according to the rule $((a)^(x))\cdot ((b)^(x))=((\left(a\cdot b \right))^(x))$, and in the latter simply multiplied the number 100 by a fraction.
Now note that the numbers on the left (at the base) and on the right are somewhat similar. How? Yes, it’s obvious: they are powers of the same number! We have:
\[\begin(align)& \frac(1000)(27)=\frac(((10)^(3)))(((3)^(3)))=((\left(\frac( 10)(3) \right))^(3)); \\& \frac(9)(100)=\frac(((3)^(2)))(((10)^(3)))=((\left(\frac(3)(10) \right))^(2)). \\\end(align)\]
Thus, our equation will be rewritten as follows:
\[((\left(((\left(\frac(10)(3) \right))^(3)) \right))^(x-1))=((\left(\frac(3 )(10)\right))^(2))\]
\[((\left(((\left(\frac(10)(3) \right))^(3)) \right))^(x-1))=((\left(\frac(10 )(3) \right))^(3\left(x-1 \right)))=((\left(\frac(10)(3) \right))^(3x-3))\]
In this case, on the right you can also get a degree with the same base, for which it is enough to simply “turn over” the fraction:
\[((\left(\frac(3)(10) \right))^(2))=((\left(\frac(10)(3) \right))^(-2))\]
Our equation will finally take the form:
\[\begin(align)& ((\left(\frac(10)(3) \right))^(3x-3))=((\left(\frac(10)(3) \right)) ^(-2)); \\& 3x-3=-2; \\& 3x=1; \\& x=\frac(1)(3). \\\end(align)\]
That's the solution. His main idea boils down to the fact that even with different bases we try, by hook or by crook, to reduce these bases to the same thing. Elementary transformations of equations and rules for working with powers help us with this.
But what rules and when to use? How do you understand that in one equation you need to divide both sides by something, and in another you need to factor the base of the exponential function?
The answer to this question will come with experience. Try your hand at simple equations first, and then gradually complicate the problems - and very soon your skills will be enough to solve any exponential equation from the same Unified State Exam or any independent/test work.
And to help you in this difficult task, I suggest downloading a set of equations from my website for solving it yourself. All equations have answers, so you can always test yourself.
Belgorod State University
DEPARTMENT algebra, number theory and geometry
Topic: Exponential power equations and inequalities.
Thesis student of the Faculty of Physics and Mathematics
Scientific supervisor:
______________________________
Reviewer: _______________________________
________________________
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 suggestions. | ||
| List of used literature. | |||
| 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 matter in which amazingly intertwine and mathematical science, and pedagogy, and didactics, and psychology, and even philosophy.
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” is now not a search for mythical optimal system teaching “everyone and everything”, but the child himself. But then - of necessity - the teacher.
In the school course of algebra and began analysis, grades 10 - 11, with passing the Unified State Exam per 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 technique of solving them, it seems to me, is very useful: it increases mental and creativity students, completely new horizons are opening 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, and independence, which will be useful to them in later life. And also there is repetition, expansion and deep assimilation of educational material.
Work on this topic diploma research I started 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.
An analysis of students' written examination papers shows that the lack of coverage of the question of the negative value of the argument of an 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 topic , 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 a complete analysis of the solution of exponential 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 works devoted to the use of various methods 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 suggestions.
List of used literature.
Chapter I analyzes the literature
