1. An exponential function is a function of the form y(x) = a x, depending on the exponent x, with a constant value of the base of the degree a, where a > 0, a ≠ 0, xϵR (R is the set of real numbers).
Let's consider graph of the function if the base does not satisfy the condition: a>0
a) a< 0
If a< 0 – возможно возведение в целую степень или в рациональную степень с нечетным показателем.
a = -2
If a = 0, the function y = is defined and has a constant value of 0
c) a =1
If a = 1, the function y = is defined and has a constant value of 1
Function Domain (DOF) Range of permissible function values (APV) 3. Zeros of the function (y = 0) 4. Points of intersection with the ordinate axis oy (x = 0) 5. Increasing, decreasing functions If , then the function f(x) increases 6. Even, odd function The function y = is not symmetrical with respect to the 0y axis and with respect to the origin of coordinates, therefore it is neither even nor odd. (General function) 7. The function y = has no extrema 8. Properties of a degree with a real exponent: Let a > 0; a≠1 Then for xϵR; yϵR: Properties of degree monotonicity: if , then If a> 0, then . 9. Relative position of the function The larger the base a, the closer to the axes x and oy a > 1, a = 20 Example 1. Hypermarket of knowledge >>Mathematics >>Mathematics 10th grade >> Exponential function, its properties and graph Let's consider the expression 2x and find its values for various rational values of the variable x, for example, for x = 2; In general, no matter what rational meaning we assign to the variable x, we can always calculate the corresponding numerical value of the expression 2 x. Thus, we can talk about exponential functions y=2 x, defined on the set Q of rational numbers: Let's look at some properties of this function. Property 1.- increasing function. We carry out the proof in two stages. On the left side of the last inequality we have , and on the right side 1. This means that the last inequality can be rewritten in the form Second phase. Let x 1 and x 2 be numbers, and x 1 and x 2< х2. Составим разность 2 х2 -2 х1 и выполним некоторые ее преобразования: (we denoted the difference x 2 - x 1 with the letter r). Since r is a positive rational number, then by what was proven at the first stage, 2 r > 1, i.e. 2 r -1 >0. The number 2x" is also positive, which means that the product 2 x-1 (2 Г -1) is also positive. Thus, we have proven that inequality 2 Xg -2x" >0. So, from the inequality x 1< х 2 следует, что 2х" <2 x2 , а это и означает, что функция у -2х - возрастающая. Property 2. limited from below and not limited from above. That this function is not of the greatest importance is obvious, since, as we have just seen, it is not bounded above. But it is limited from below, why doesn’t it have a minimum value? Let's assume that 2 r is the smallest value of the function (r is some rational indicator). Let's take a rational number q<г. Тогда в силу возрастания функции у=2 х будем иметь 2 x <2г. А это значит, что 2 r не может служить наименьшим значением функции. All this is good, you say, but why do we consider the function y-2 x only on the set of rational numbers, why don’t we consider it like other known functions on the entire number line or on some continuous interval of the number line? What's stopping us? Let's think about the situation. The number line contains not only rational, but also ir rational numbers. For the previously studied functions this did not bother us. For example, we found the values of the function y = x2 equally easily for both rational and irrational values of x: it was enough to square the given value of x. But with the function y=2 x the situation is more complicated. If the argument x is given a rational meaning, then in principle x can be calculated (go back again to the beginning of the paragraph, where we did exactly this). What if argument x is given an irrational meaning? How, for example, to calculate? We don't know this yet. It is known that 1; 1,7; 1,73; 1,732; 1,7320; 1,73205; 1,732050; 1,7320508;... . It is clear that 1.732 = 1.7320, and 1.732050 = 1.73205. To avoid such repetitions, we discard those members of the sequence that end with the number 0. Then we get an increasing sequence: 1; 1,7; 1,73; 1,732; 1,73205; 1,7320508;... . Accordingly, the sequence increases All terms of this sequence are positive numbers less than 22, i.e. this sequence is limited. According to Weierstrass' theorem (see § 30), if a sequence is increasing and bounded, then it converges. In addition, from § 30 we know that if a sequence converges, it does so only to one limit. It was agreed that this single limit should be considered the value of a numerical expression. And it doesn’t matter that it is very difficult to find even an approximate value of the numerical expression 2; it is important that this is a specific number (after all, we were not afraid to say that, for example, it is the root of a rational equation, Let's mark the points on the coordinate plane (Fig. 194), they mark a certain line, let's draw it (Fig. 195). Rigorous proofs of the listed properties of the function y-2 x are given in the course of higher mathematics. We discussed some of these properties to one degree or another earlier, some of them are clearly demonstrated by the constructed graph (see Fig. 195). For example, the lack of parity or oddness of a function is geometrically related to the lack of symmetry of the graph, respectively, relative to the y-axis or relative to the origin. Any function of the form y = a x, where a > 1, has similar properties. In Fig. 196 in one coordinate system were constructed, graphs of functions y=2 x, y=3 x, y=5 x. Let's now consider the function and create a table of values for it: 1) Definition. A function of the form is called an exponential function. The graph of the function y=a x for a> 1 is shown in Fig. 201, and for 0<а < 1 - на рис. 202. The curve shown in Fig. 201 or 202 is called exponent. In fact, mathematicians usually call the exponential function itself y = a x. So the term "exponent" is used in two senses: both to name the exponential function and to name the graph of the exponential function. Usually the meaning is clear whether we are talking about an exponential function or its graph. Pay attention to the geometric feature of the graph of the exponential function y=ax: the x-axis is the horizontal asymptote of the graph. True, this statement is usually clarified as follows. In other words These are examples of power functions; In general, y = x r, where r is a specific number, is a power function (the argument x is contained in the base of the degree); An "exotic" function like y = x" is considered neither exponential nor power (it is sometimes called exponential). Second important note. Usually one does not consider an exponential function with base a = 1 or with base a satisfying the inequality a<0 (вы, конечно, помните, что выше, в определении показательной функции, оговорены условия: а >0 and a The fact is that if a = 1, then for any value of x the equality Ix = 1 holds. Thus, the exponential function y = a" with a = 1 "degenerates" into a constant function y = 1 - this is not interesting. If a = 0, then 0x = 0 for any positive value of x, i.e. we get the function y = 0, defined for x > 0 - this is also uninteresting. If, finally, a<0, то выражение а" имеет смысл лишь при целых значениях х, а мы все-таки предпочитаем рассматривать функции, определенные на сплошных промежутках. Before moving on to solving the examples, note that the exponential function is significantly different from all the functions you have studied so far. To thoroughly study a new object, you need to consider it from different angles, in different situations, so there will be many examples. Solution, a) Having constructed graphs of the functions y = 2 x and y = 1 in one coordinate system, we notice (Fig. 203) that they have one common point (0; 1). This means that the equation 2x = 1 has a single root x =0. So, from the equation 2x = 2° we get x = 0. b) Having constructed graphs of the functions y = 2 x and y = 4 in one coordinate system, we notice (Fig. 203) that they have one common point (2; 4). This means that the equation 2x = 4 has a single root x = 2. So, from the equation 2 x = 2 2 we get x = 2. c) and d) Based on the same considerations, we conclude that the equation 2 x = 8 has a single root, and to find it, graphs of the corresponding functions do not need to be built; it is clear that x = 3, since 2 3 = 8. Similarly, we find the only root of the equation Solution. You can proceed like this: build a graph of the y-3 x function, then stretch it from the x axis by a factor of 3, and then raise the resulting graph up by 2 scale units. But it is more convenient to use the fact that 3- 3* = 3 * + 1, and, therefore, build a graph of the function y = 3 x * 1 + 2. Let's move on, as we have done many times in such cases, to an auxiliary coordinate system with the origin at the point (-1; 2) - dotted lines x = - 1 and 1x = 2 in Fig. 207. Let’s “link” the function y=3* to the new coordinate system. To do this, select control points for the function Example 4. Solve equation and inequalities: Solution, a) Let us construct graphs of the functions y=5* and y=6-x in one coordinate system (Fig. 208). They intersect at one point; judging by the drawing, this is point (1; 5). The check shows that in fact the point (1; 5) satisfies both the equation y = 5* and the equation y = 6-x. The abscissa of this point serves as the only root of the given equation. So, the equation 5 x = 6 - x has a single root x = 1. b) and c) The exponent y-5x lies above the straight line y=6-x, if x>1, this is clearly visible in Fig. 208. This means that the solution to the inequality 5*>6's can be written as follows: x>1. And the solution to the inequality 5x<6 - х можно записать так: х < 1. Example 5. Given a function Exponential function
Function of the form y = a x , where a is greater than zero and a is not equal to one is called an exponential function. Basic properties of the exponential function: 1. The domain of definition of the exponential function will be the set of real numbers. 2. The range of values of the exponential function will be the set of all positive real numbers. Sometimes this set is denoted as R+ for brevity. 3. If in an exponential function the base a is greater than one, then the function will be increasing over the entire domain of definition. If in the exponential function for the base a the following condition is satisfied 0 4. All basic properties of degrees will be valid. The main properties of degrees are represented by the following equalities: a x *a y = a (x+y) ;
(a x )/(a y ) = a (x-y) ;
(a*b) x = (a x )*(a y );
(a/b) x = a x /b x ;
(a x )
y = a (x * y) .
These equalities will be valid for all real values of x and y. 5. The graph of an exponential function always passes through the point with coordinates (0;1) 6. Depending on whether the exponential function increases or decreases, its graph will have one of two forms. The following figure shows a graph of an increasing exponential function: a>0. The following figure shows the graph of a decreasing exponential function: 0 Both the graph of an increasing exponential function and the graph of a decreasing exponential function, according to the property described in the fifth paragraph, pass through the point (0;1). 7. An exponential function does not have extremum points, that is, in other words, it does not have minimum and maximum points of the function. If we consider a function on any specific segment, then the function will take on the minimum and maximum values at the ends of this interval. 8. The function is not even or odd. An exponential function is a function general view. This can be seen from the graphs; none of them are symmetrical either with respect to the Oy axis or with respect to the origin of coordinates. Logarithm
Logarithms have always been considered a difficult topic in school mathematics courses. There are many different definitions logarithm, but for some reason most textbooks use the most complex and unsuccessful of them. We will define the logarithm simply and clearly. To do this, let's create a table: So, we have powers of two. If you take the number from the bottom line, you can easily find the power to which you will have to raise two to get this number. For example, to get 16, you need to raise two to the fourth power. And to get 64, you need to raise two to the sixth power. This can be seen from the table. And now - actually, the definition of the logarithm: Definition Logarithm to base a of argument x is the power to which the number must be raised a to get the number x. Designation log a x = b For example, 2 3 = 8 ⇒ log 2 8 = 3 (the base 2 logarithm of 8 is three because 2 3 = 8). With the same success, log 2 64 = 6, since 2 6 = 64. The operation of finding the logarithm of a number to a given base is calledlogarithm
. So, let's add a new line to our table: Unfortunately, not all logarithms are calculated so easily. For example, try to find log 2 5. The number 5 is not in the table, but logic dictates that the logarithm will lie somewhere on the interval. Because 2 2< 5 < 2 3 , а чем больше степень двойки, тем больше получится число.
Such numbers are called irrational: the numbers after the decimal point can be written ad infinitum, and they are never repeated. If the logarithm turns out to be irrational, it is better to leave it that way: log 2 5, log 3 8, log 5 100. It is important to understand that a logarithm is an expression with two variables (the base and the argument). At first, many people confuse where the basis is and where the argument is. To avoid annoying misunderstandings, just look at the picture: Before us is nothing more than the definition of a logarithm. Remember: logarithm is a power , into which the base must be built in order to obtain an argument. It is the base that is raised to a power - it is highlighted in red in the picture. It turns out that the base is always at the bottom! I tell my students this wonderful rule at the very first lesson - and no confusion arises. We've figured out the definition - all that remains is to learn how to count logarithms, i.e. get rid of the "log" sign. To begin with, we note that From the definition two things follow important facts:
The argument and the base must always be greater than zero. This follows from the definition of a degree by a rational exponent, to which the definition of a logarithm is reduced. The base must be different from one, since one to any degree still remains one. Because of this, the question “to what power must one be raised to get two” is meaningless. There is no such degree! Such restrictions are called range of acceptable values(ODZ). It turns out that the logarithm’s ODZ looks like this: log a x = b ⇒
x > 0, a > 0, a ≠ 1. Please note that no restrictions on number b (logarithm value) does not overlap. For example, the logarithm may well be negative: log 2 0.5 = −1, because 0.5 = 2 −1. However, now we are considering only numerical expressions, where it is not required to know the VA of the logarithm. All restrictions have already been taken into account by the authors of the problems. But when logarithmic equations and inequalities come into play, DL requirements will become mandatory. After all, the basis and argument may contain very strong constructions that do not necessarily correspond to the above restrictions. Now consider the general scheme for calculating logarithms. It consists of three steps: Provide a reason a and argument x in the form of a power with the minimum possible base greater than one. Along the way, it’s better to get rid of decimals; Solve with respect to a variable b equation: x = a b ; The resulting number b will be the answer. That's all! If the logarithm turns out to be irrational, this will be visible already in the first step. The requirement that the base be greater than one is very important: this reduces the likelihood of error and greatly simplifies the calculations. Same with decimals: if you immediately convert them to regular ones, there will be many fewer errors. Let's see how this scheme works on specific examples:
Calculate the logarithm: log 5 25 Let's imagine the base and argument as a power of five: 5 = 5 1 ; 25 = 5 2 ; Let's create and solve the equation: We received the answer: 2. Calculate the logarithm: Let's imagine the base and argument as a power of three: 3 = 3 1 ; 1/81 = 81 −1 = (3 4) −1 = 3 −4 ; Let's create and solve the equation: We received the answer: −4. −4
Calculate the logarithm: log 4 64 Let's imagine the base and argument as a power of two: 4 = 2 2 ; 64 = 2 6 ; Let's create and solve the equation: We received the answer: 3. Calculate the logarithm: log 16 1 Let's imagine the base and argument as a power of two: 16 = 2 4 ; 1 = 2 0 ; Let's create and solve the equation: We received the answer: 0. Calculate the logarithm: log 7 14 Let's imagine the base and argument as a power of seven: 7 = 7 1 ; 14 cannot be represented as a power of seven, since 7 1< 14 < 7 2 ; From the previous paragraph it follows that the logarithm does not count; The answer is no change: log 7 14. log 7 14 A small note on the last example. How can you be sure that a number is not an exact power of another number? It’s very simple - just factor it into prime factors. If the expansion has at least two different factors, the number is not an exact power. Find out whether the numbers are exact powers: 8; 48; 81; 35; 14. 8 = 2 · 2 · 2 = 2 3 - exact degree, because there is only one multiplier; 8, 81 - exact degree; 48, 35, 14 - no. Note also that the prime numbers themselves are always exact powers of themselves. Decimal logarithm
Some logarithms are so common that they have a special name and symbol. Definition Decimal logarithm from argument x is the logarithm to base 10, i.e. the power to which the number 10 must be raised to get the number x. Designation lg x For example, log 10 = 1; lg 100 = 2; lg 1000 = 3 - etc. From now on, when a phrase like “Find lg 0.01” appears in a textbook, know that this is not a typo. This is a decimal logarithm. However, if you are unfamiliar with this notation, you can always rewrite it: Everything that is true for ordinary logarithms is also true for decimal logarithms. Natural logarithm
There is another logarithm that has its own designation. In some ways, it's even more important than decimal. It's about about the natural logarithm. Definition Natural logarithm from argument x is the logarithm to the base e , i.e. the power to which a number must be raised e to get the number x. Designation ln x Many people will ask: what is the number e? This is an irrational number; its exact value cannot be found and written down. I will give only the first figures: We will not go into detail about what this number is and why it is needed. Just remember that e - base of natural logarithm: Thus ln e = 1; ln e 2 = 2; ln e 16 = 16 - etc. On the other hand, ln 2 is an irrational number. In general, the natural logarithm of any rational number is irrational. Except, of course, for one: ln 1 = 0. For natural logarithms, all the rules that are true for ordinary logarithms are valid. Basic properties of logarithms Logarithms, like any numbers, can be added, subtracted and transformed in every way. But since logarithms are not exactly ordinary numbers, they have their own rules, which are called basic properties. You definitely need to know these rules - without them not a single serious logarithmic problem can be solved. In addition, there are very few of them - you can learn everything in one day. So let's get started. Adding and subtracting logarithms
Consider two logarithms with the same bases: log a x and log a y . Then they can be added and subtracted, and: log a x + log a y =log a (
x ·
y );
log a x − log a y =log a (
x :
y ).
So, the sum of logarithms is equal to the logarithm of the product, and the difference is equal to the logarithm of the quotient. Note: key moment here are the same reasons. If the reasons are different, these rules do not work! These formulas will help you calculate a logarithmic expression even when its individual parts are not considered (see lesson "
"). Take a look at the examples and see: Find the value of the expression: log 6 4 + log 6 9. Since logarithms have the same bases, we use the sum formula: Find the value of the expression: log 2 48 − log 2 3. The bases are the same, we use the difference formula: Find the value of the expression: log 3 135 − log 3 5. Again the bases are the same, so we have: As you can see, the original expressions are made up of “bad” logarithms, which are not calculated separately. But after the transformations, completely normal numbers are obtained. Many are built on this fact test papers. Yes, test-like expressions are offered in all seriousness (sometimes with virtually no changes) on the Unified State Examination. Extracting the exponent from the logarithm
Now let's complicate the task a little. What if the base or argument of a logarithm is a power? Then the exponent of this degree can be taken out of the sign of the logarithm according to the following rules: It is easy to see that the last rule follows the first two. But it’s better to remember it anyway - in some cases it will significantly reduce the amount of calculations. Of course All these rules make sense if the ODZ of the logarithm is observed: a > 0, a ≠ 1, x > 0. And one more thing: learn to apply all formulas not only from left to right, but also vice versa, i.e. You can enter the numbers before the logarithm sign into the logarithm itself. This is what is most often required. Find the value of the expression: log 7 49 6 . Let's get rid of the degree in the argument using the first formula: Find the meaning of the expression: Note that the denominator contains a logarithm, the base and argument of which are exact powers: 16 = 2 4 ; 49 = 7 2. We have: I think the last example requires some clarification. Where have logarithms gone? Until the very last moment we work only with the denominator. We presented the base and argument of the logarithm standing there in the form of powers and took out the exponents - we got a “three-story” fraction. Now let's look at the main fraction. The numerator and denominator contain the same number: log 2 7. Since log 2 7 ≠ 0, we can reduce the fraction - 2/4 will remain in the denominator. According to the rules of arithmetic, the four can be transferred to the numerator, which is what was done. The result was the answer: 2. Transition to a new foundation
Speaking about the rules for adding and subtracting logarithms, I specifically emphasized that they only work with the same bases. What if the reasons are different? What if they are not exact powers of the same number? Formulas for transition to a new foundation come to the rescue. Let us formulate them in the form of a theorem: Theorem Let the logarithm log be given a x . Then for any number c such that c > 0 and c ≠ 1, the equality is true: In particular, if we put c = x, we get: From the second formula it follows that the base and argument of the logarithm can be swapped, but in this case the entire expression is “turned over”, i.e. the logarithm appears in the denominator.
These formulas are rarely found in ordinary numerical expressions. It is possible to evaluate how convenient they are only by deciding logarithmic equations and inequalities. However, there are problems that cannot be solved at all except by moving to a new foundation. Let's look at a couple of these: Find the value of the expression: log 5 16 log 2 25. Note that the arguments of both logarithms contain exact powers. Let's take out the indicators: log 5 16 = log 5 2 4 = 4log 5 2; log 2 25 = log 2 5 2 = 2log 2 5; Now let’s “reverse” the second logarithm: Since the product does not change when rearranging factors, we calmly multiplied four and two, and then dealt with logarithms. Find the value of the expression: log 9 100 lg 3. The base and argument of the first logarithm are exact powers. Let's write this down and get rid of the indicators: Now let's get rid of the decimal logarithm by moving to a new base: Basic logarithmic identity
Often in the solution process it is necessary to represent a number as a logarithm to a given base. In this case, the following formulas will help us: In the first case, the number n becomes an indicator of the degree standing in the argument. Number n can be absolutely anything, because it’s just a logarithm value. The second formula is actually a paraphrased definition. This is what it's called:basic logarithmic identity.
In fact, what happens if the number b is raised to such a power that the number b to this power gives the number a? That's right: the result is the same number a. Read this paragraph carefully again - many people get stuck on it. Like formulas for moving to a new base, the basic logarithmic identity is sometimes the only possible solution. Task Find the meaning of the expression: Solution Note that log 25 64 = log 5 8 - simply took the square from the base and the argument of the logarithm. Taking into account the rules for multiplying powers with the same base, we get: 200
If anyone doesn't know, this was a real task from the Unified State Exam :) Logarithmic unit and logarithmic zero
In conclusion, I will give two identities that can hardly be called properties - rather, they are consequences of the definition of the logarithm. They constantly appear in problems and, surprisingly, create problems even for “advanced” students. log a a = 1 is logarithmic unit. Remember once and for all: logarithm to any base a from this very base is equal to one. log a 1 = 0 is logarithmic zero. Base a can be anything, but if the argument contains one, the logarithm is equal to zero! Because a 0 = 1 is a direct consequence of the definition. That's all the properties. Be sure to practice putting them into practice! EXPONENTARY AND LOGARITHMIC FUNCTIONS VIII § 179 Basic properties of the exponential function In this section we will study the basic properties of the exponential function y = a
x
(1) Let us remember that under A
in formula (1) we mean any fixed positive number other than 1. Property 1. The domain of an exponential function is the set of all real numbers.
In fact, with a positive A
expression A
x
defined for any real number X
. Property 2. The exponential function accepts only positive values.
Indeed, if X
> 0, then, as was proven in § 176, A
x
> 0. If X
<. 0, то A
x
= Where - X
already more than zero. That's why A -
x
> 0. But then A
x
= > 0. Finally, when X
= 0 A
x
= 1. The 2nd property of the exponential function has a simple graphical interpretation. It lies in the fact that the graph of this function (see Fig. 246 and 247) is located entirely above the abscissa axis. Property 3. If
A
>1, then when
X
> 0 A
x
> 1, and when
X
< 0 A
x
< 1. If
A
< 1, тoh, on the contrary, when
X
> 0 A
x
< 1, and when
X
< 0 A
x
> 1. This property of the exponential function also allows for a simple geometric interpretation. At A
> 1 (Fig. 246) curves y = a
x
located above the straight line at
= 1 at X
> 0 and below straight line at
= 1 at X
< 0. If A
< 1 (рис. 247), то, наоборот, кривые y = a
x
located below the straight line at
= 1 at X
> 0 and above this straight line at X
< 0. Let us give a rigorous proof of the 3rd property. Let A
> 1 and X
- an arbitrary positive number. Let's show that A
x
> 1. If the number X
rational ( X
= m /
n
) , That A
x
= A
m/
n
= n
√a
m
. Because the A
> 1, then A
m
> 1, But the root of a number greater than one is obviously also greater than 1. If X
is irrational, then there are positive rational numbers X"
And X"
, which serve as decimal approximations of a number x : X"< х < х"
. But then, by definition of a degree with an irrational exponent A
x"
< A
x
< A
x""
. As shown above, the number A
x"
more than one. Therefore the number A
x
, greater than A
x"
, must also be greater than 1, So, we have shown that when a
>1 and arbitrary positive X
A
x
> 1. If the number X
was negative, then we would have A
x
= where the number is X
would already be positive. That's why A -
x
> 1. Therefore, A
x
= < 1. Thus, when A
> 1 and arbitrary negative x
A
x
< 1. The case when 0< A
< 1, легко сводится к уже рассмотренному случаю. Учащимся предлагается убедиться в этом самостоятельно. Property 4. If x
= 0, then regardless of a
A
x
=1. This follows from the definition of degree zero; the zero power of any number other than zero is equal to 1. Graphically, this property is expressed in the fact that for any A
curve at
= A
x
(see Fig. 246 and 247) intersects the axis at
at the point with ordinate 1. Property 5. At
A
>1 exponential function
= A
x
is monotonically increasing, and for a
< 1 - monotonically decreasing.
This property also allows for a simple geometric interpretation. At A
> 1 (Fig. 246) curve at
= A
x
with growth X
rises higher and higher, and when A
< 1 (рис. 247) - опускается все ниже и ниже. Let us give a rigorous proof of the 5th property. Let A
> 1 and X
2 > X
1 . Let's show that A
x
2
> A
x
1
Because the X
2 > X
1 ., then X
2 = X
1 + d
, Where d
- some positive number. That's why A
x
2
- A
x
1
= A
x
1
+
d
- A
x
1
= A
x
1
(A
d
- 1) By the 2nd property of the exponential function A
x
1 > 0. Since d
> 0, then by the 3rd property of the exponential function A
d
> 1. Both factors in the product A
x
1
(A
d
- 1) are positive, therefore this product itself is positive. Means, A
x
2
- A
x
1 > 0, or A
x
2
> A
x
1, which is what needed to be proven. So, when a
> 1 function at
= A
x
is monotonically increasing. Similarly, it is proved that when A
< 1 функция at
= A
x
is monotonically decreasing. Consequence. If two powers of the same positive number other than 1 are equal, then their exponents are equal.
In other words, if A
b
= A
c
(A
> 0 and A
=/= 1), b = c
. Indeed, if the numbers b
And With
were not equal, then due to the monotonicity of the function at
= A
x
the greater of them would correspond to A
>1 greater, and when A
< 1 меньшее значение этой функции. Таким образом, было бы или A
b
> A
c
, or A
b
< A
c
. Both contradict the condition A
b
= A
c
. It remains to admit that b = c
. Property 6. If a
> 1, then with an unlimited increase in the argument
X
(X
-> ∞
) function values
at
= A
x
also grow indefinitely
(at
-> ∞
). When the argument decreases without limit
X
(X
-> -∞
) the values of this function tend to zero while remaining positive
(at->0; at
> 0). Taking into account the monotonicity of the function proved above at
= A
x
, we can say that in the case under consideration the function at
= A
x
monotonically increases from 0 to ∞
. If
0 <A
< 1, then with an unlimited increase in the argument x (x -> ∞), the values of the function y = a x tend to zero, while remaining positive
(at->0; at
> 0). When the argument x decreases without limit
(X
-> -∞
) the values of this function grow unlimitedly
(at
-> ∞
).
Due to the monotonicity of the function y = a x
we can say that in this case the function at
= A
x
monotonically decreases from ∞
to 0. The 6th property of the exponential function is clearly reflected in Figures 246 and 247. We will not strictly prove it. All we have to do is establish the range of variation of the exponential function y = a x
(A
> 0, A
=/= 1). Above we proved that the function y = a x
takes only positive values and either increases monotonically from 0 to ∞
(at A
> 1), or decreases monotonically from ∞
to 0 (at 0< A
<. 1). Однако остался невыясненным следующий вопрос: не претерпевает ли функция y = a x
Are there any jumps when you change? Does it take any positive values? This issue is resolved positively. If A
> 0 and A
=/= 1, then whatever the positive number is at
0 will definitely be found X
0 , such that A
x
0
= at
0 . (Due to the monotonicity of the function y = a x
specified value X
0 will, of course, be the only one.) Proving this fact is beyond the scope of our program. Its geometric interpretation is that for any positive value at
0 function graph y = a x
will definitely intersect with a straight line at
= at
0 and, moreover, only at one point (Fig. 248). From this we can draw the following conclusion, which we formulate as property 7. Property 7. The area of change of the exponential function y = a x
(A
> 0, A
=/= 1)is the set of all positive numbers.
Exercises
1368. Find the domains of definition of the following functions: 1369. Which of these numbers is greater than 1 and which is less than 1: 1370. Based on what property of the exponential function can it be stated that a) (5 / 7) 2.6 > (5 / 7) 2.5; b) (4 / 3) 1.3 > (4 / 3) 1.2 1371. Which number is greater: A) π -
√3 or (1/ π
) -
√3 ; c) (2 / 3) 1 +
√6 or (2 / 3) √2 +
√5
; b) ( π /
4) 1 +
√3 or ( π /
4) 2; d) (√3) √2 -
√5 or (√3) √3 -
2 ? 1372. Are the inequalities equivalent: 1373. What can be said about numbers X
And at
, If a x
= and y
, Where A
- a given positive number? 1374. 1) Is it possible among all the values of the function at
= 2x
highlight: 2) Is it possible among all the values of the function at
= 2 |
x|
highlight: A) highest value; b) the smallest value? Solving most mathematical problems in one way or another involves transforming numerical, algebraic or functional expressions. The above applies especially to the decision. In the versions of the Unified State Exam in mathematics, this type of problem includes, in particular, task C3. Learning to solve C3 tasks is important not only for the purpose of successful completion Unified State Examination, but also for the reason that this skill will be useful when studying a mathematics course in higher school. When completing tasks C3, you have to decide different kinds equations and inequalities. Among them are rational, irrational, exponential, logarithmic, trigonometric, containing modules (absolute values), as well as combined ones. This article discusses the main types of exponential equations and inequalities, as well as various methods their decisions. Read about solving other types of equations and inequalities in the “” section in articles devoted to methods for solving C3 problems from Unified State Exam options mathematics. Before we begin to analyze specific exponential equations and inequalities, as a math tutor, I suggest you brush up on some theoretical material, which we will need. Function of the form y = a x, Where a> 0 and a≠ 1 is called exponential function. Basic properties of exponential function y = a x: The graph of the exponential function is exponent: Graphs of exponential functions (exponents) Indicative are called equations in which the unknown variable is found only in exponents of some powers. For solutions exponential equations you need to know and be able to use the following simple theorem: Theorem 1. Exponential equation a f(x) = a g(x) (Where a > 0, a≠ 1) is equivalent to the equation f(x) = g(x). In addition, it is useful to remember the basic formulas and operations with degrees: Title="Rendered by QuickLaTeX.com">!} Example 1. Solve the equation: Solution: We use the above formulas and substitution: The equation then becomes: Discriminant of the received quadratic equation positive: Title="Rendered by QuickLaTeX.com">!} This means that this equation has two roots. We find them: Moving on to reverse substitution, we get: The second equation has no roots, since the exponential function is strictly positive throughout the entire domain of definition. Let's solve the second one: Taking into account what was said in Theorem 1, we move on to the equivalent equation: x= 3. This will be the answer to the task. Answer: x = 3. Example 2. Solve the equation: Solution: The equation has no restrictions on the range of permissible values, since the radical expression makes sense for any value x(exponential function y = 9 4 -x positive and not equal to zero). We solve the equation by equivalent transformations using the rules of multiplication and division of powers: The last transition was carried out in accordance with Theorem 1. Answer:x= 6.
Example 3. Solve the equation: Solution: both sides of the original equation can be divided by 0.2 x. This transition will be equivalent, since this expression is greater than zero for any value x(the exponential function is strictly positive in its domain of definition). Then the equation takes the form: Answer: x = 0. Example 4. Solve the equation: Solution: we simplify the equation to an elementary one by means of equivalent transformations using the rules of division and multiplication of powers given at the beginning of the article: Dividing both sides of the equation by 4 x, as in the previous example, is an equivalent transformation, since this expression is not equal to zero for any values x. Answer: x = 0. Example 5. Solve the equation: Solution: function y = 3x, standing on the left side of the equation, is increasing. Function y = —x The -2/3 on the right side of the equation is decreasing. This means that if the graphs of these functions intersect, then at most one point. IN in this case it is not difficult to guess that the graphs intersect at the point x= -1. There will be no other roots. Answer: x = -1. Example 6. Solve the equation: Solution: we simplify the equation by means of equivalent transformations, keeping in mind everywhere that the exponential function is strictly greater than zero for any value x and using the rules for calculating the product and quotient of powers given at the beginning of the article: Answer: x = 2. Indicative are called inequalities in which the unknown variable is contained only in exponents of some powers. For solutions exponential inequalities knowledge of the following theorem is required: Theorem 2. If a> 1, then the inequality a f(x) > a g(x) is equivalent to an inequality of the same meaning: f(x) > g(x). If 0< a < 1, то показательное неравенство a f(x) > a g(x) is equivalent to an inequality with the opposite meaning: f(x) < g(x). Example 7. Solve the inequality: Solution: Let's present the original inequality in the form: Let's divide both sides of this inequality by 3 2 x, in this case (due to the positivity of the function y= 3 2x) the inequality sign will not change: Let's use the substitution: Then the inequality will take the form: So, the solution to the inequality is the interval: moving to the reverse substitution, we get: Due to the positivity of the exponential function, the left inequality is satisfied automatically. Using the well-known property of the logarithm, we move on to the equivalent inequality: Since the base of the degree is a number greater than one, equivalent (by Theorem 2) is the transition to the following inequality: So, we finally get answer: Example 8. Solve the inequality: Solution: Using the properties of multiplication and division of powers, we rewrite the inequality in the form: Let's introduce a new variable: Taking this substitution into account, the inequality takes the form: Multiplying the numerator and denominator of the fraction by 7, we obtain the following equivalent inequality: So, the following values of the variable satisfy the inequality t: Then, moving to the reverse substitution, we get: Since the base of the degree here is greater than one, the transition to the inequality will be equivalent (by Theorem 2): Finally we get answer: Example 9. Solve the inequality: Solution: We divide both sides of the inequality by the expression: It is always greater than zero (due to the positivity of the exponential function), so there is no need to change the inequality sign. We get: t located in the interval: Moving on to the reverse substitution, we find that the original inequality splits into two cases: The first inequality has no solutions due to the positivity of the exponential function. Let's solve the second one: Example 10. Solve the inequality: Solution: Parabola branches y = 2x+2-x 2 are directed downwards, therefore it is limited from above by the value that it reaches at its vertex: Parabola branches y = x 2 -2x The +2 in the indicator are directed upward, which means it is limited from below by the value that it reaches at its vertex: At the same time, the function also turns out to be bounded from below y = 3 x 2 -2x+2, which is on the right side of the equation. It reaches its smallest value at the same point as the parabola in the exponent, and this value is 3 1 = 3. So, the original inequality can only be true if the function on the left and the function on the right take on the value , equal to 3 (the intersection of the ranges of values of these functions is only this number). This condition is satisfied at a single point x = 1. Answer: x= 1. In order to learn to decide exponential equations and inequalities it is necessary to constantly train in solving them. Various things can help you with this difficult task. methodological manuals, problem books on elementary mathematics, collections of competitive problems, mathematics classes at school, as well as individual lessons with a professional tutor. I sincerely wish you success in your preparation and excellent results in the exam. P.S. Dear guests! Please do not write requests to solve your equations in the comments. Unfortunately, I have absolutely no time for this. Such messages will be deleted. Please read the article. Perhaps in it you will find answers to questions that did not allow you to solve your task on your own.
2. Let's take a closer look at the exponential function:
0
If , then the function f(x) decreases
Function y= , at 0
This follows from the properties of monotonicity of a power with a real exponent.
b> 0; b≠1
For example:
The exponential function is continuous at any point ϵ R.
If a0, then the exponential function takes a form close to y = 0.
If a1, then further from the ox and oy axes and the graph takes on a form close to the function y = 1.
Construct a graph of y =
First stage. Let us prove that if r is a positive rational number, then 2 r >1.
Two cases are possible: 1) r - natural number, r = n; 2) ordinary irreducible fraction, 
So, in any case, the inequality 2 r > 1 holds, which is what needed to be proved.
The boundedness of the function from below follows from the inequality 2 x >0, which is valid for any values of x from the domain of definition of the function. At the same time, no matter what positive number M you take, you can always choose an exponent x such that the inequality 2 x >M will be satisfied - which characterizes the unboundedness of the function from above. Let us give a number of examples. 
Property 3. has neither the smallest nor the largest value.
Mathematicians have found a way out; that's how they reasoned.
Consider a sequence of rational numbers - decimal approximations of a number by disadvantage:
the root of a trigonometric equation, without really thinking about what exactly these numbers are: 
So, we have found out what meaning mathematicians put into the symbol 2^. Similarly, you can determine what and in general what a a is, where a is an irrational number and a > 1.
But what if 0<а <1? Как вычислить, например, ? Самым естественным способом: считать, что свести вычисления к случаю, когда основание степени больше 1.
Now we can talk not only about powers with arbitrary rational exponents, but also about powers with arbitrary real exponents. It has been proven that degrees with any real exponents have all the usual properties of degrees: when multiplying powers with the same bases, the exponents are added, when dividing, they are subtracted, when raising a degree to a power, they are multiplied, etc. But the most important thing is that now we can talk about the function y-ax defined on the set of all real numbers.
Let's return to the function y = 2 x and construct its graph. To do this, let’s create a table of function values y=2x:
Properties of the function y - 2 x:
1)
2) is neither even nor odd; 248
3) increases;
5) has neither the largest nor the smallest values;
6) continuous;
7)
8) convex downwards.
Let's mark the points on the coordinate plane (Fig. 197), they mark a certain line, let's draw it (Fig. 198).
Function Properties
2) is neither even nor odd;
3) decreases;
4) not limited from above, limited from below;
5) there is neither the largest nor the smallest value;
6) continuous;
7)
8) convex downwards.
Any function of the form y = a x has similar properties, where O<а <1. На рис. 200 в одной системе координат построены графики функций 
Please note: function graphs 
those. y=2 x, symmetrical about the y-axis (Fig. 201). This is a consequence of the general statement (see § 13): the graphs of the functions y = f(x) and y = f(-x) are symmetrical about the y-axis. Similarly, the graphs of the functions y = 3 x and
To summarize what has been said, we will give a definition of the exponential function and highlight its most important properties.
Basic properties of the exponential function y = a x
The x-axis is the horizontal asymptote of the graph of the function
First important note. Schoolchildren often confuse the terms: power function, exponential function. Compare:
These are examples of exponential functions.
y = a", where a is a specific number (positive and different from 1), is an exponential function (the argument x is contained in the exponent).
Example 1. 
So, from the equation 2x = 2 3 we got x = 3, and from the equation 2 x = 2 x we got x = -4.
e) The graph of the function y = 2 x is located above the graph of the function y = 1 for x > 0 - this is clearly readable in Fig. 203. This means that the solution to the inequality 2x > 1 is the interval
f) The graph of the function y = 2 x is located below the graph of the function y = 4 at x<2 - это хорошо читается по рис. 203. Значит, решением неравенства 2х <4служит промежуток
You probably noticed that the basis for all the conclusions made when solving example 1 was the property of monotonicity (increase) of the function y = 2 x. Similar reasoning allows us to verify the validity of the following two theorems. 
, but we will build them not in the old, but in the new coordinate system (these points are marked in Fig. 207). Then we will construct an exponent from the points - this will be the required graph (see Fig. 207).
To find the largest and smallest values of a given function on the segment [-2, 2], we take advantage of the fact that the given function is increasing, and therefore it takes its smallest and largest values, respectively, at the left and right ends of the segment.
So:
Answer: a)x = 1; b)x>1; c)x<1.
Prove that 
Solution. According to the condition We have.
where a is the base, x is the argument, b - actually, what the logarithm is equal to.
log 5 25 = b ⇒ (5 1) b = 5 2 ⇒ 5 b = 5 2 ⇒ b = 2;
log 4 64 = b ⇒ (2 2) b = 2 6 ⇒ 2 2 b = 2 6 ⇒ 2b = 6 ⇒ b = 3;
log 16 1 = b ⇒ (2 4) b = 2 0 ⇒ 2 4 b = 2 0 ⇒ 4b = 0 ⇒ b = 0;
48 = 6 · 8 = 3 · 2 · 2 · 2 · 2 = 3 · 2 4 - is not an exact power, since there are two factors: 3 and 2;
81 = 9 · 9 = 3 · 3 · 3 · 3 = 3 4 - exact degree;
35 = 7 · 5 - again not an exact power;
14 = 7 · 2 - again not an exact degree;
log x = log 10 x
e = 2.718281828459...
ln x = log e x
log 6 4 + log 6 9 = log 6 (4 9) = log 6 36 = 2.
log 2 48 − log 2 3 = log 2 (48: 3) = log 2 16 = 4.
log 3 135 − log 3 5 = log 3 (135: 5) = log 3 27 = 3.
log 7 49 6 = 6 log 7 49 = 6 2 = 12
Exponential function
What is an exponential function?
Graph of an Exponential Function
Solving exponential equations
Solving exponential inequalities
Sergey Valerievich
