Provides reference data on the exponential function - basic properties, graphs and formulas. The following issues are considered: domain of definition, set of values, monotonicity, inverse function, derivative, integral, expansion in power series and representation using complex numbers.
Definition
Exponential function
is a generalization of the product of n numbers equal to a:
y (n) = a n = a·a·a···a,
to the set of real numbers x:
y (x) = ax.
Here a is a fixed real number, which is called basis of the exponential function.
An exponential function with base a is also called exponent to base a.
The generalization is carried out as follows.
For natural x = 1, 2, 3,...
, the exponential function is the product of x factors:
.
Moreover, it has properties (1.5-8) (), which follow from the rules for multiplying numbers. For zero and negative values of integers, the exponential function is determined using formulas (1.9-10). For fractional values x = m/n rational numbers, , it is determined by formula (1.11). For real , the exponential function is defined as the limit of the sequence:
,
where is an arbitrary sequence of rational numbers converging to x: .
With this definition, the exponential function is defined for all , and satisfies properties (1.5-8), as for natural x.
A rigorous mathematical formulation of the definition of an exponential function and the proof of its properties is given on the page “Definition and proof of the properties of an exponential function”.
Properties of the Exponential Function
The exponential function y = a x has the following properties on the set of real numbers ():
(1.1)
defined and continuous, for , for all ;
(1.2)
for a ≠ 1
has many meanings;
(1.3)
strictly increases at , strictly decreases at ,
is constant at ;
(1.4)
at ;
at ;
(1.5)
;
(1.6)
;
(1.7)
;
(1.8)
;
(1.9)
;
(1.10)
;
(1.11)
,
.
Other useful formulas.
.
Formula for converting to an exponential function with a different exponent base:
When b = e, we obtain the expression of the exponential function through the exponential:
Private values
, , , , .
The figure shows graphs of the exponential function
y (x) = ax
for four values degree bases: a = 2
, a = 8
, a = 1/2
and a = 1/8
. It can be seen that for a > 1
the exponential function increases monotonically. The larger the base of the degree a, the more strong growth. At 0
< a < 1
the exponential function decreases monotonically. The smaller the exponent a, the stronger the decrease.
Ascending, descending
The exponential function for is strictly monotonic and therefore has no extrema. Its main properties are presented in the table.
| y = a x , a > 1 | y = ax, 0 < a < 1 | |
| Domain | - ∞ < x < + ∞ | - ∞ < x < + ∞ |
| Range of values | 0 < y < + ∞ | 0 < y < + ∞ |
| Monotone | monotonically increases | monotonically decreases |
| Zeros, y = 0 | No | No |
| Intercept points with the ordinate axis, x = 0 | y = 1 | y = 1 |
| + ∞ | 0 | |
| 0 | + ∞ |
Inverse function
The inverse of an exponential function with base a is the logarithm to base a.
If , then
.
If , then
.
Differentiation of an exponential function
To differentiate an exponential function, its base must be reduced to the number e, apply the table of derivatives and the rule for differentiating a complex function.
To do this you need to use the property of logarithms
and the formula from the derivatives table:
.
Let an exponential function be given:
.
We bring it to the base e:
Let's apply the rule of differentiation of complex functions. To do this, introduce the variable
Then
From the table of derivatives we have (replace the variable x with z):
.
Since is a constant, the derivative of z with respect to x is equal to
.
According to the rule of differentiation of a complex function:
.
Derivative of an exponential function
.
Derivative of nth order:
.
Deriving formulas > > >
An example of differentiating an exponential function
Find the derivative of a function
y = 3 5 x
Solution
Let's express the base of the exponential function through the number e.
3 = e ln 3
Then
.
Enter a variable
.
Then
From the table of derivatives we find:
.
Because the 5ln 3 is a constant, then the derivative of z with respect to x is equal to:
.
According to the rule of differentiation of a complex function, we have:
.
Answer
Integral
Expressions using complex numbers
Consider the function complex number z:
f (z) = a z
where z = x + iy; i 2 = - 1
.
Let us express the complex constant a in terms of modulus r and argument φ:
a = r e i φ
Then
.
The argument φ is not uniquely defined. IN general view
φ = φ 0 + 2 πn,
where n is an integer. Therefore the function f (z) is also not clear. Its main significance is often considered
.
Series expansion
.
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
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 irrational 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
Lesson No.2
Topic: Exponential function, its properties and graph. Target: Check the quality of mastering the concept of “exponential function”; to develop skills in recognizing the exponential function, using its properties and graphs, teaching students to use analytical and graphical forms of recording the exponential function; provide a working environment in the classroom. Equipment: board, posters Lesson form: class lesson Lesson type: practical lesson Lesson type: lesson in teaching skills and abilities Lesson Plan 1. Organizational moment 2. Independent work and checking homework 3. Problem solving 4. Summing up 5. Homework During the classes. 1. Organizational moment
: Hello. Open your notebooks, write down today’s date and the topic of the lesson “Exponential Function”. Today we will continue to study the exponential function, its properties and graph. 2. Independent work and checking homework
.
Target: check the quality of mastery of the concept of “exponential function” and check the completion of the theoretical part of the homework Method: test task, frontal survey As homework, you were given numbers from the problem book and a paragraph from the textbook. We won’t check your execution of numbers from the textbook now, but you will hand in your notebooks at the end of the lesson. Now the theory will be tested in the form of a small test. The task is the same for everyone: you are given a list of functions, you must find out which of them are indicative (underline them). And next to the exponential function you need to write whether it is increasing or decreasing. Option 1 Answer B)
D) - exponential, decreasing Option 2 Answer D) - exponential, decreasing D)
- exponential, increasing Option 3 Answer A)
- exponential, increasing B)
- exponential, decreasing Option 4 Answer A)
- exponential, decreasing IN)
- exponential, increasing Now let’s remember together which function is called exponential? A function of the form , where and , is called an exponential function. What is the scope of this function? All real numbers. What is the range of the exponential function? All positive real numbers. Decreases if the base of the power is greater than zero but less than one. In what case does an exponential function decrease in its domain of definition? Increasing if the base of the power is greater than one. 3. Problem solving
Target: to develop skills in recognizing an exponential function, using its properties and graphs, teach students to use analytical and graphical forms of writing an exponential function Method: demonstration by the teacher of solving typical problems, oral work, work at the blackboard, work in a notebook, conversation between the teacher and students. The properties of the exponential function can be used when comparing 2 or more numbers. For example: No. 000. Compare the values and if a) - IN) - G) - - No. 000. Compare the numbers: a) and Therefore, the function increases, then Why ? Increasing function and Therefore, the function is decreasing, then Both functions increase throughout their entire domain of definition, since they are exponential with a base of power greater than one. What is the meaning behind it? We build graphs: Which function increases faster when striving https://pandia.ru/text/80/379/images/image062_0.gif" width="20 height=25" height="25"> Which function decreases faster when striving https://pandia.ru/text/80/379/images/image062_0.gif" width="20 height=25" height="25"> On the interval, which of the functions has greater value at a specific point? D), https://pandia.ru/text/80/379/images/image068_0.gif" width="69" height="57 src=">. First, let's find out the scope of definition of these functions. Do they coincide? Yes, the domain of these functions is all real numbers. Name the scope of each of these functions. The ranges of these functions coincide: all positive real numbers. Determine the type of monotonicity of each function. All three functions decrease throughout their entire domain of definition, since they are exponential with a base of powers less than one and greater than zero. What special point exists in the graph of an exponential function? What is the meaning behind it? Whatever the basis of the degree of an exponential function, if the exponent contains 0, then the value of this function is 1. We build graphs: Let's analyze the graphs. How many points of intersection do the graphs of functions have? Which function decreases faster when trying https://pandia.ru/text/80/379/images/image070.gif" width="41 height=57" height="57"> Which function increases faster when striving https://pandia.ru/text/80/379/images/image070.gif" width="41 height=57" height="57"> On the interval, which of the functions has greater value at a specific point? On the interval, which of the functions has greater value at a specific point? Why are exponential functions with for different reasons have only one intersection point? Exponential functions are strictly monotonic throughout their entire domain of definition, so they can intersect only at one point. The next task will focus on using this property. No. 000. Find the largest and smallest values of the given function on the given interval a) . Recall that a strictly monotonic function takes its minimum and maximum values at the ends of a given segment. And if the function is increasing, then its highest value will be at the right end of the segment, and the smallest at the left end of the segment (demonstration on the poster, using the example of an exponential function). If the function is decreasing, then its largest value will be at the left end of the segment, and the smallest at the right end of the segment (demonstration on the poster, using the example of an exponential function). The function is increasing, because, therefore, the smallest value of the function will be at the point https://pandia.ru/text/80/379/images/image075_0.gif" width="145" height="29">. Points b ) Students solve the task in their notebooks Decreasing function Decreasing function Increasing function - No. 000. Find the largest and smallest value of the given function on the given interval a) 
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 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 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 greatest 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.
..gif" width="37" height="20 src=">, then this is a rather complicated job: we would have to take the cube root of 3 and 9, and compare them. But we know that it increases, this in its own way turn means that as the argument increases, the value of the function increases, that is, we just need to compare the values of the argument and , it is obvious that 
(can be demonstrated on a poster showing an increasing exponential function). And always, when solving such examples, you first determine the base of the exponential function, compare it with 1, determine monotonicity and proceed to compare the arguments. In the case of a decreasing function: when the argument increases, the value of the function decreases, therefore, we change the sign of inequality when moving from inequality of arguments to inequality of functions. Next, we solve orally: b) 
, V) 
d) solve the notebooks yourself, we will check them orally.
greatest value of the function on the segment
the smallest value of a function on a segment
the smallest value of a function on a segment
greatest value of the function on the segment
. This task is almost the same as the previous one. But what is given here is not a segment, but a ray. We know that the function is increasing, and it has neither the largest nor the smallest value on the entire number line https://pandia.ru/text/80/379/images/image063_0.gif" width="68" height ="20">, and tends to at , i.e. on the ray the function at tends to 0, but does not have its smallest value, but it has the largest value at the point 
. Points b) 
, V) 
, G) 
Solve the notebooks yourself, we will check them orally.
