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. At zero and negative values 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 ones, the exponential function is defined as sequence limit:
,
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 differentiation rule 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 complex number function 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
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.
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 check 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) 
..gif" width="37" height="20 src=">, then this is quite a difficult job: we would have to extract cube root from 3 and from 9, and compare them. But we know that it increases, this in 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) 
- 
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 higher 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 striving 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 do exponential functions with different bases 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 value given function on a 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 ) 
, V) 
d) solve the notebooks yourself, we will check them orally.
Students solve the task in their notebooks
|
Decreasing function
|
Decreasing function
|
Increasing function
|
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- No. 000. Find the largest and smallest value of the given function on the given interval a) 
. 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 own lowest value, but it has the greatest value at the point 
. Points b) 
, V) 
, G) 
Solve the notebooks yourself, we will check them orally.
Majority decision mathematical problems is somehow related to the transformation of numerical, algebraic or functional expressions. The above applies especially to the decision. In the versions of the Unified State Examination 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 passing the Unified State Exam, but also for the reason that this skill will be useful when studying a mathematics course in high 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 for solving them. 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 that we will need.
Exponential function
What is an exponential function?
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:
Graph of an Exponential Function
The graph of the exponential function is exponent:
Graphs of exponential functions (exponents)
Solving exponential equations
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:
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Example 1. Solve the equation:
Solution: We use the above formulas and substitution:
The equation then becomes:
Discriminant of the received quadratic equation positive:
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It means that given 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.
Solving exponential inequalities
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 of 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 inequality is satisfied following values variable 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 in elementary mathematics, collections of competitive problems, mathematics classes at school, as well as individual sessions with a professional tutor. I sincerely wish you success in your preparation and excellent results in the exam.
Sergey Valerievich
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.
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 course mathematics. There are many different definitions of 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
where a is the base, x is the argument, b - actually, what the logarithm is equal to.
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 to our table new line:
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 more degree twos, the larger the number.
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:
log 5 25 = b ⇒ (5 1) b = 5 2 ⇒ 5 b = 5 2 ⇒ b = 2;
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:
log 4 64 = b ⇒ (2 2) b = 2 6 ⇒ 2 2 b = 2 6 ⇒ 2b = 6 ⇒ b = 3;
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:
log 16 1 = b ⇒ (2 4) b = 2 0 ⇒ 2 4 b = 2 0 ⇒ 4b = 0 ⇒ b = 0;
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 break it down 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;
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;
8, 81 - exact degree; 48, 35, 14 - no.
Let us also note that we ourselves prime numbers are always exact degrees 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:
log x = log 10 x
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. We are talking 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 impossible to find and record. I will give only the first figures:
e = 2.718281828459...
We will not go into detail about what this number is and why it is needed. Just remember that e - base of natural logarithm:
ln x = log e x
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. Please note: the key point here is the same grounds. If the reasons are different, these rules do not work!
These formulas will help you calculate logarithmic expression even when its individual parts are not counted (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:
log 6 4 + log 6 9 = log 6 (4 9) = log 6 36 = 2.
Find the value of the expression: log 2 48 − log 2 3.
The bases are the same, we use the difference formula:
log 2 48 − log 2 3 = log 2 (48: 3) = log 2 16 = 4.
Find the value of the expression: log 3 135 − log 3 5.
Again the bases are the same, so we have:
log 3 135 − log 3 5 = log 3 (135: 5) = log 3 27 = 3.
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:
log 7 49 6 = 6 log 7 49 = 6 2 = 12
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 conventional 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. Considering the rules for multiplying powers with the same basis, 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 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 a 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) the greatest value; b) the smallest value?