Concentration of attention:
Definition. Function species is called exponential function .
Comment. Base exclusion a numbers 0; 1 and negative values a explained by the following circumstances:
The analytic expression itself a x in these cases, it retains its meaning and can be encountered in solving problems. For example, for the expression x y dot x = 1; y = 1 included in the area allowed values.
Construct graphs of functions: and .
 |
 |
| Graph of an exponential function | |
| y= a x, a > 1 | y= a x , 0< a < 1 |
 |
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Properties of the exponential function
| Properties of the exponential function | y= a x, a > 1 | y= a x , 0< a < 1 |
|
||
| 2. Range of function values | ||
| 3. Intervals of comparison with the unit | at x> 0, a x > 1 | at x > 0, 0< a x < 1 |
| at x < 0, 0< a x < 1 | at x < 0, a x > 1 | |
| 4. Even, odd. | The function is neither even nor odd (function general view). | |
| 5. Monotony. | increases monotonically by R | decreases monotonically by R |
| 6. Extremes. | The exponential function has no extrema. | |
| 7.Asymptote | Axis O x is a horizontal asymptote. | |
| 8. For any real values x And y; |  |
|
When the table is filled, tasks are solved in parallel with the filling.
Task number 1. (To find the domain of the function).
What argument values are valid for functions:
Task number 2. (To find the range of the function).
The figure shows a graph of a function. Specify the scope and scope of the function:
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 |
 |
 |
Task number 3. (To indicate the intervals of comparison with the unit).
Compare each of the following powers with one:
Task number 4. (To study the function for monotonicity).
Compare real numbers by magnitude m And n if:
Task number 5. (To study the function for monotonicity).
Make a conclusion about the basis a, if:
y(x) = 10 x ; f(x) = 6 x ; z(x) - 4x
How are the graphs of exponential functions relative to each other for x > 0, x = 0, x< 0?
One coordinate plane function graphs are built:
y(x) = (0,1) x ; f(x) = (0.5) x ; z(x) = (0.8) x .
How are the graphs of exponential functions relative to each other for x > 0, x = 0, x< 0?
| Number
one of the most important constants in mathematics. By definition, it equal to the limit of the sequence
with unlimited
increasing n
. Designation e introduced Leonard Euler
in 1736. He calculated the first 23 digits of this number in decimal notation, and the number itself was named after Napier "the non-peer number."
Number e plays a special role in mathematical analysis. Exponential function with base e, called the exponent and denoted y = e x. First signs numbers e easy to remember: two, a comma, seven, the year of Leo Tolstoy's birth - two times, forty-five, ninety, forty-five. |
 |
Homework:
Kolmogorov p. 35; No. 445-447; 451; 453.
Repeat the algorithm for constructing graphs of functions containing a variable under the module sign.
Knowledge basic elementary functions, their properties and graphs no less important than knowing the multiplication table. They are like a foundation, everything is based on them, everything is built from them, and everything comes down to them.
In this article, we list all the main elementary functions, give their graphs and give them without derivation and proofs. properties of basic elementary functions according to the scheme:
- behavior of the function on the boundaries of the domain of definition, vertical asymptotes (if necessary, see the article classification of breakpoints of a function);
- even and odd;
- convexity (convexity upwards) and concavity (convexity downwards) intervals, inflection points (if necessary, see the article function convexity, convexity direction, inflection points, convexity and inflection conditions);
- oblique and horizontal asymptotes;
- singular points of functions;
- special properties of some functions (for example, the smallest positive period for trigonometric functions).
If you are interested in or, then you can go to these sections of the theory.
Basic elementary functions are: constant function (constant), root of the nth degree, power function, exponential, logarithmic function, trigonometric and inverse trigonometric functions.
Page navigation.
Permanent function.
A constant function is given on the set of all real numbers by the formula , where C is some real number. The constant function assigns to each real value of the independent variable x the same value of the dependent variable y - the value С. A constant function is also called a constant.
The graph of a constant function is a straight line parallel to the x-axis and passing through a point with coordinates (0,C) . For example, let's show graphs of constant functions y=5 , y=-2 and , which in the figure below correspond to the black, red and blue lines, respectively.
Properties of a constant function.
- Domain of definition: the whole set of real numbers.
- The constant function is even.
- Range of values: set consisting of singular FROM .
- A constant function is non-increasing and non-decreasing (that's why it is constant).
- It makes no sense to talk about the convexity and concavity of the constant.
- There is no asymptote.
- The function passes through the point (0,C) of the coordinate plane.
The root of the nth degree.
Consider the basic elementary function, which is given by the formula , where n is natural number, greater than one.
The root of the nth degree, n is an even number.
Let's start with the nth root function for even values of the root exponent n .
For example, we give a picture with images of graphs of functions 
and , they correspond to black, red and blue lines.
The graphs of the functions of the root of an even degree have a similar form for other values of the indicator.
Properties of the root of the nth degree for even n .
The root of the nth degree, n is an odd number.
The root function of the nth degree with an odd exponent of the root n is defined on the entire set of real numbers. For example, we present graphs of functions 
and , the black, red, and blue curves correspond to them.
For other odd values of the root exponent, the graphs of the function will have a similar appearance.
Properties of the root of the nth degree for odd n .
Power function.
The power function is given by a formula of the form .
Consider the type of graphs power function and properties of the power function depending on the value of the exponent.
Let's start with a power function with an integer exponent a . In this case, the form of graphs of power functions and the properties of functions depend on the even or odd exponent, as well as on its sign. Therefore, we first consider power functions for odd positive values of the exponent a , then for even positive ones, then for odd negative exponents, and finally, for even negative a .
The properties of power functions with fractional and irrational exponents (as well as the type of graphs of such power functions) depend on the value of the exponent a. We will consider them, firstly, when a is from zero to one, secondly, when a is greater than one, thirdly, when a is from minus one to zero, and fourthly, when a is less than minus one.
In conclusion of this subsection, for the sake of completeness, we describe a power function with zero exponent.
Power function with odd positive exponent.
Consider a power function with an odd positive exponent, that is, with a=1,3,5,… .
The figure below shows graphs of power functions - black line, - blue line, - red line, - green line. For a=1 we have linear function y=x .
Properties of a power function with an odd positive exponent.
Power function with even positive exponent.
Consider a power function with an even positive exponent, that is, for a=2,4,6,… .
As an example, let's take graphs of power functions - black line, - blue line, - red line. For a=2 we have quadratic function, whose graph is quadratic parabola.
Properties of a power function with an even positive exponent.
Power function with an odd negative exponent.
Look at the plots of the power function for odd negative values exponent, that is, when a=-1,-3,-5,… .
The figure shows graphs of exponential functions as examples - black line, - blue line, - red line, - green line. For a=-1 we have inverse proportionality, whose graph is hyperbola.
Properties of a power function with an odd negative exponent.
Power function with an even negative exponent.
Let's move on to the power function at a=-2,-4,-6,….
The figure shows graphs of power functions - black line, - blue line, - red line.
Properties of a power function with an even negative exponent.
A power function with a rational or irrational exponent whose value is greater than zero and less than one.
Note! If a is a positive fraction with an odd denominator, then some authors consider the interval to be the domain of the power function. At the same time, it is stipulated that the exponent a is an irreducible fraction. Now the authors of many textbooks on algebra and the beginnings of analysis DO NOT DEFINE power functions with an exponent in the form of a fraction with an odd denominator for negative values of the argument. We will adhere to just such a view, that is, we will consider the domains of power functions with fractional positive exponents to be the set . We encourage students to get your teacher's perspective on this subtle point to avoid disagreement.
Consider a power function with rational or irrational exponent a , and .
We present graphs of power functions for a=11/12 (black line), a=5/7 (red line), (blue line), a=2/5 (green line).
A power function with a non-integer rational or irrational exponent greater than one.
Consider a power function with a non-integer rational or irrational exponent a , and .
Let us present the graphs of the power functions given by the formulas 
(black, red, blue and green lines respectively).
For other values of the exponent a , the graphs of the function will have a similar look.
Power function properties for .
A power function with a real exponent that is greater than minus one and less than zero.
Note! If a is a negative fraction with an odd denominator, then some authors consider the interval 
. At the same time, it is stipulated that the exponent a is an irreducible fraction. Now the authors of many textbooks on algebra and the beginnings of analysis DO NOT DEFINE power functions with an exponent in the form of a fraction with an odd denominator for negative values of the argument. We will adhere to just such a view, that is, we will consider the domains of power functions with fractional fractional negative exponents to be the set, respectively. We encourage students to get your teacher's perspective on this subtle point to avoid disagreement.
We pass to the power function , where .
In order to have a good idea of the type of graphs of power functions for , we give examples of graphs of functions 
(black, red, blue, and green curves, respectively).
Properties of a power function with exponent a , .
A power function with a non-integer real exponent that is less than minus one.
Let us give examples of graphs of power functions for 
, they are depicted in black, red, blue and green lines, respectively.
Properties of a power function with a non-integer negative exponent less than minus one.
When a=0 and we have a function - this is a straight line from which the point (0; 1) is excluded (the expression 0 0 was agreed not to attach any importance).
Exponential function.
One of the main elementary functions is exponential function.
Graph of the exponential function, where and takes a different form depending on the value of the base a. Let's figure it out.
First, consider the case when the base of the exponential function takes a value from zero to one, that is, .
For example, we present the graphs of the exponential function for a = 1/2 - the blue line, a = 5/6 - the red line. The graphs of the exponential function have a similar appearance for other values of the base from the interval .
Properties of an exponential function with a base less than one.
We turn to the case when the base of the exponential function is greater than one, that is, .
As an illustration, we present graphs of exponential functions - the blue line and - the red line. For other values of the base, greater than one, the graphs of the exponential function will have a similar appearance.
Properties of an exponential function with a base greater than one.
Logarithmic function.
next main elementary function is the logarithmic function , where , . The logarithmic function is defined only for positive values of the argument, that is, for .
Schedule logarithmic function takes a different form depending on the value of the base a.
Majority Decision math problems somehow connected with the transformation of numerical, algebraic or functional expressions. This applies especially to the solution. In the USE variants in mathematics, this type of task includes, in particular, task C3. Learning how to solve C3 tasks is important not only for the purpose of successful passing the exam, but also for the reason that this skill is useful when studying a mathematics course in higher education.
Performing tasks C3, you have to solve various types of 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 under the heading "" in articles devoted to methods for solving C3 problems from USE options mathematics.
Before proceeding to the analysis of specific exponential equations and inequalities, as a math tutor, I suggest you brush up on some of the theoretical material that we will need.
Exponential function
What is an exponential function?
View function y = a x, where a> 0 and a≠ 1, called exponential function.
Main exponential function properties y = a x:
Graph of an exponential function
The graph of the exponential function is exhibitor:
Graphs of exponential functions (exponents)
Solution of exponential equations
indicative called equations in which the unknown variable is found only in exponents of any 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 actions with degrees:
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Example 1 Solve the equation:
Solution: use the above formulas and substitution:
The equation then becomes:
Received discriminant quadratic equation positive:
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It means that given equation has two roots. We find them:
Going back to substitution, we get:
The second equation has no roots, since the exponential function is strictly positive over the entire domain of definition. Let's solve the second one:
Taking into account what was said in Theorem 1, we pass 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 area of admissible 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 on its domain). Then the equation takes the form:
Answer: x = 0.
Example 4 Solve the equation:
Solution: we simplify the equation to an elementary one by 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 given expression not equal to zero for any value 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-2/3, standing on the right side of the equation, is decreasing. This means that if the graphs of these functions intersect, then at most at one point. In this case, it is easy 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 equivalent transformations, bearing 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 partial powers given at the beginning of the article:
Answer: x = 2.
Solving exponential inequalities
indicative called inequalities in which the unknown variable is contained only in the 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 exponential inequality 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: represent the original inequality in the form:
Divide both sides of this inequality by 3 2 x, and (due to the positiveness of the function y= 3 2x) the inequality sign will not change:
Let's use a substitution:
Then the inequality takes the form:
So, the solution to the inequality is the interval:
passing to the reverse substitution, we get:
The left inequality, due to the positiveness of the exponential function, is fulfilled automatically. Using the well-known property of the logarithm, we pass to the equivalent inequality:
Since the base of the degree is a number greater than one, equivalent (by Theorem 2) will be 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:
With this substitution, the inequality takes the form:
Multiply the numerator and denominator of the fraction by 7, we get the following equivalent inequality:
So, the inequality is satisfied by the following values of the variable t:
Then, going back to substitution, we get:
Since the base of the degree here is greater than one, it is equivalent (by Theorem 2) to pass to the inequality:
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 (because the exponential function is positive), so the inequality sign does not need to be changed. We get:
t , which are in the interval:
Passing 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, hence it is bounded from above by the value it reaches at its vertex:
Parabola branches y = x 2 -2x+2, which is in the indicator, are directed upwards, which means it is limited from below by the value that it reaches at its top:
At the same time, the function turns out to be bounded from below y = 3 x 2 -2x+2 on the right side of the equation. She reaches her the 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 the value 3 at one point (by the intersection the ranges of these functions is only this number). This condition is satisfied at a single point x = 1.
Answer: x= 1.
To learn how to solve exponential equations and inequalities, you need to constantly train in their solution. In this difficult matter, various teaching aids, 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 brilliant results in the exam.
Sergey Valerievich
P.S. Dear guests! Please do not write requests for solving your equations in the comments. Unfortunately, I don't have time for this at all. 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.
1. An exponential function is a function of the form y(x) \u003d 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).
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 rational degree with an odd score.
a = -2
If a = 0 - the function y = is defined and has a constant value 0
c) a \u003d 1
If a = 1 - the function y = is defined and has a constant value of 1
Function domain (OOF) Area of allowable function values (ODZ) 3. Zeros of the function (y = 0) 4. Points of intersection with the y-axis (x = 0) 5. Increasing, decreasing function If , then the function f(x) increases 6. Even, odd functions The function y = is not symmetrical about the 0y axis and about the origin, therefore it is neither even nor odd. (general function) 7. The function y \u003d has no extremums 8. Properties of a degree with a real exponent: Let a > 0; a≠1 Then for xϵR; yϵR: Degree monotonicity properties: if , then If a> 0, then . 9. Relative location of the function The larger the base a, the closer to the x and y axes a > 1, a = 20 Example 1 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. The main properties of the exponential function: 1. The domain of the exponential function will be the set of real numbers. 2. The range 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 we have next condition 0
4. All the 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 the 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 types. The following figure shows a graph of an increasing exponential function: a>0. The following figure is a graph of a decreasing exponential function: 0 Both the graph of the increasing exponential function and the graph of the 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 the function on any particular segment, then the function will take the minimum and maximum values at the ends of this interval. 8. The function is not even or odd. An exponential function is a general function. This can also be seen from the graphs, none of them is symmetrical either about the Oy axis or about the origin. Logarithm
Logarithms have always been considered difficult topic in school course mathematics. 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. Let's create a table for this: So, we have powers of two. If you take the number from the bottom line, then you can easily find the power to which you have to raise a 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 - in fact, the definition of the logarithm: Definition Logarithm base a from 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). Might as well 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 row to our table: Unfortunately, not all logarithms are considered 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 segment. 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 indefinitely, and they never repeat. If the logarithm turns out to be irrational, it is better to leave it like this: log 2 5, log 3 8, log 5 100. It is important to understand that the logarithm is an expression with two variables (base and argument). At first, many people confuse where the base is and where the argument is. To avoid annoying misunderstandings, just take a look at the picture: Before us is nothing more than the definition of the logarithm. Remember: the logarithm is a power , to which you need to raise the base to get the argument. It is the base that is raised to a power - in the picture it is highlighted in red. It turns out that the base is always at the bottom! I tell this wonderful rule to my students at the very first lesson - and there is no confusion. We figured out the definition - it remains to learn how to count logarithms, i.e. get rid of the "log" sign. To begin with, we note that Two important facts follow from the definition: The argument and base must always be greater than zero. This follows from the definition of the degree by a rational exponent, to which the definition of the logarithm is reduced. The base must be different from unity, since a unit to any power is still a unit. Because of this, the question “to what power must one be raised to get two” is meaningless. There is no such degree! Such restrictions called valid range(ODZ). It turns out that the ODZ of the logarithm looks like this: log a x = b ⇒
x > 0, a > 0, a ≠ 1. Notice that no limit on the 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 ODZ of the logarithm. All restrictions have already been taken into account by the compilers of the problems. But when logarithmic equations and inequalities come into play, the DHS requirements will become mandatory. Indeed, in the basis and argument there can be very strong constructions, which do not necessarily correspond to the above restrictions. Now consider the general scheme for calculating logarithms. It consists of three steps: Submit Foundation a and argument x as a power with the smallest possible base greater than one. Along the way, it is better to get rid of decimal fractions; Decide on a Variable b equation: x = a b ; Received number b will be the answer. That's all! If the logarithm turns out to be irrational, this will be seen already at the first step. The requirement that the base be greater than one is very relevant: this reduces the likelihood of error and greatly simplifies calculations. Similar to decimals: if you immediately translate them into ordinary ones, there will be many times less errors. Let's see how this scheme works with specific examples: Calculate the logarithm: log 5 25 Let's represent the base and the argument as a power of five: 5 = 5 1 ; 25 = 52; Let's make and solve the equation: Received an answer: 2. Calculate the logarithm: Let's represent the base and the argument as a power of three: 3 = 3 1 ; 1/81 \u003d 81 -1 \u003d (3 4) -1 \u003d 3 -4; Let's make and solve the equation: Got the answer: -4. −4
Calculate the logarithm: log 4 64 Let's represent the base and the argument as a power of two: 4 = 2 2 ; 64 = 26; Let's make and solve the equation: Received an answer: 3. Calculate the logarithm: log 16 1 Let's represent the base and the argument as a power of two: 16 = 2 4 ; 1 = 20; Let's make and solve the equation: Received a response: 0. Calculate the logarithm: log 7 14 Let's represent the base and the argument as a power of seven: 7 = 7 1 ; 14 is not represented as a power of seven, because 7 1< 14 < 7 2 ; It follows from the previous paragraph that the logarithm is not considered; The answer is no change: log 7 14. log 7 14 A small note on the last example. How to make sure that a number is not an exact power of another number? Very simple - just decompose it into prime factors. If the decomposition contains at least two different multiplier, the number is not an exact power. Find out if the exact powers of the number are: 8; 48; 81; 35; fourteen. 8 \u003d 2 2 2 \u003d 2 3 - the 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 designation. Definition Decimal logarithm from argument x is the logarithm to base 10, i.e. the power to which you need to raise the number 10 to get the number x. Designation lg x For example, log 10 = 1; log 100 = 2; lg 1000 = 3 - etc. From now on, when a phrase like “Find lg 0.01” appears in the textbook, know that this is not a typo. This is the decimal logarithm. However, if you are not used to such a designation, you can always rewrite it: Everything that is true for ordinary logarithms is also true for decimals. natural logarithm
There is another logarithm that has its own notation. In a sense, it is even more important than decimal. It's about about the natural logarithm. Definition natural logarithm from argument x is the base logarithm e , i.e. the power to which the number must be raised e to get the number x. Designation ln x Many will ask: what is the number e? This is an irrational number, its exact value cannot be found and written down. Here are just the first numbers: We will not delve into what this number is and why it is needed. Just remember that e is the base of the natural logarithm: Thus ln e = 1; log 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 irrational. Except, of course, unity: ln 1 = 0. For natural logarithms all the rules that are true for ordinary logarithms are valid. Basic properties of logarithms Logarithms, like any number, can be added, subtracted and converted in every possible way. But since logarithms are not quite ordinary numbers, there are rules here, which are called basic properties. These rules must be known - not a single serious problem is solved without them. logarithmic problem. In addition, there are very few of them - everything can be learned in one day. So let's get started. Addition and subtraction of logarithms
Consider two logarithms with the same base: 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 the logarithms is equal to the logarithm of the product, and the difference is the logarithm of the quotient. Please note: the key point here is the same bases. If the bases are different, these rules do not work! These formulas will help you calculate the logarithmic expression even when its individual parts are not considered (see the lesson "
"). Take a look at the examples - and see: Find the value of the expression: log 6 4 + log 6 9. Since the bases of logarithms are the same, 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 considered separately. But after transformations quite normal numbers turn out. Based on this fact, many test papers. Yes, that control - similar expressions in all seriousness (sometimes - with virtually no changes) are offered at the exam. Removing the exponent from the logarithm
Now let's complicate the task a little. What if there is a degree in the base or argument of the logarithm? 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 their 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 logarithm is observed: a > 0, a ≠ 1, x > 0 you can enter the numbers before the sign of the logarithm 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 according to the first formula: Find the value of the expression: Note that the denominator is a logarithm whose base and argument are exact powers: 16 = 2 4 ; 49 = 72. We have: I think the last example needs clarification. Where have logarithms gone? Until the very last moment, we work only with the denominator. They presented the base and the argument of the logarithm standing there in the form of degrees and took out the indicators - they got a “three-story” fraction. Now let's look at the main fraction. The numerator and denominator have 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 was done. The result is 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 bases are different? What if they are not exact powers of the same number? Formulas for transition to a new base come to the rescue. We formulate them in the form of a theorem: Theorem Let the logarithm log 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: It follows from the second formula that it is possible to interchange the base and the argument of the logarithm, but in this case the whole expression is “turned over”, i.e. the logarithm is in the denominator.
These formulas are rarely found in ordinary numerical expressions. It is possible to evaluate how convenient they are only when deciding logarithmic equations and inequalities. However, there are tasks that cannot be solved at all except by moving to a new foundation. Let's consider a couple of these: Find the value of the expression: log 5 16 log 2 25. Note that the arguments of both logarithms are exact exponents. 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 flip the second logarithm: Since the product does not change from permutation of factors, we calmly multiplied four and two, and then figured out the 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 it down and get rid of the indicators: Now let's get rid of decimal logarithm, moving to a new base: Basic logarithmic identity
Often in the process of solving it is required to represent a number as a logarithm to a given base. In this case, the formulas will help us: In the first case, the number n becomes the exponent of the argument. Number n can be absolutely anything, because it's just the value of the logarithm. The second formula is actually a paraphrased definition. It's called like this:basic logarithmic identity.
Indeed, what will happen if the number b is raised to such a degree that the number b in this degree gives the number a? That's right: this is the same number a. Read this paragraph carefully again - many people "hang" on it. Like the new base conversion formulas, the basic logarithmic identity is sometimes the only possible solution. A task Find the value of the expression: Solution Note that log 25 64 = log 5 8 - just took out the square from the base and the argument of the logarithm. Given the rules for multiplying powers with the same base, we get: 200
If someone is not in the know, this was a real task from the exam :) Logarithmic unit and logarithmic zero
In conclusion, I will give two identities that are difficult to call properties - rather, these are consequences from the definition of the logarithm. They are constantly found in problems and, surprisingly, create problems even for "advanced" students. log a a = 1 is logarithmic unit. Remember once and for all: the logarithm to any base a from this base itself is equal to one. log a 1 = 0 is logarithmic zero. Base a can be anything, but if the argument is one - the logarithm is 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!
2. Consider the exponential function in more detail:
0
If , then the function f(x) decreases
Function y= , at 0
This follows from the monotonicity properties of degree c real indicator.
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 axes x and y and the graph takes the form close to the function y \u003d 1.
Plot y=
where a is the base, x is the argument, b What exactly is the logarithm.
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 because there are two factors: 3 and 2;
81 \u003d 9 9 \u003d 3 3 3 3 \u003d 3 4 - exact degree;
35 = 7 5 - again not an exact degree;
14 \u003d 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
