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The concept of a logarithmic function

First, let's remember what a logarithm is.

Definition 1

The logarithm of a number $b\in R$ to the base $a$ ($a>0,\ a\ne 1$) is the number $c$ to which the number $a$ must be raised to obtain the number $b$.

Consider the exponential function $f\left(x\right)=a^x$, where $a >1$. This function is increasing, continuous and maps the real axis to the interval $(0,+\infty)$. Then, by the theorem on the existence of an inverse continuous function, in the set $Y=(0,+\infty)$ it has an inverse function $x=f^(-1)(y)$, which is also continuous and increases in $Y $ and maps the interval $(0,+\infty)$ to the entire real axis. This inverse function is called the logarithmic function in base $a\ (a >1)$ and is denoted $y=((log)_a x\ )$.

Now consider the exponential function $f\left(x\right)=a^x$, where $0

Thus, we have defined a logarithmic function for all possible values ​​of the base $a$. Let us consider these two cases separately.

1%24"> Function $y=((log)_a x\ ),\ a >1$

Consider properties this function.

    There are no intersections with the $Oy$ axis.

    The function is positive for $x\in (1,+\infty)$ and negative for $x\in (0,1)$

    $y"=\frac(1)(xlna)$;

    Minimum and maximum points:

    The function increases over the entire domain of definition;

    $y^("")=-\frac(1)(x^2lna)$;

    \[-\frac(1)(x^2lna)The function is convex on the entire domain of definition;

    $(\mathop(lim)_(x\to 0) y\ )=-\infty ,\ (\mathop(lim)_(x\to +\infty ) y\ )=+\infty ,\ $;

    Function graph (Fig. 1).

Figure 1. Graph of the function $y=((log)_a x\ ),\ a >1$

Function $y=((log)_a x\ ), \ 0

Consider the properties of this function.

    The domain of definition is the interval $(0,+\infty)$;

    The range of value is all real numbers;

    The function is neither even nor odd.

    Intersection points with coordinate axes:

    There are no intersections with the $Oy$ axis.

    For $y=0$, $((log)_a x\ )=0,\ x=1.$ Intersection with the $Ox$ axis: (1,0).

    The function is positive for $x\in (0,1)$ and negative for $x\in (1,+\infty)$

    $y"=\frac(1)(xlna)$;

    Minimum and maximum points:

    \[\frac(1)(xlna)=0-roots\ no\]

    There are no maximum or minimum points.

    $y^("")=-\frac(1)(x^2lna)$;

    Convexity and concavity intervals:

    \[-\frac(1)(x^2lna)>0\]

    Function graph (Fig. 2).

Examples of research and construction of logarithmic functions

Example 1

Explore and graph the function $y=2-((log)_2 x\ )$

    The domain of definition is the interval $(0,+\infty)$;

    The range of value is all real numbers;

    The function is neither even nor odd.

    Intersection points with coordinate axes:

    There are no intersections with the $Oy$ axis.

    For $y=0$, $2-((log)_2 x\ )=0,\ x=4.$ Intersection with the $Ox$ axis: (4,0).

    The function is positive for $x\in (0,4)$ and negative for $x\in (4,+\infty)$

    $y"=-\frac(1)(xln2)$;

    Minimum and maximum points:

    \[-\frac(1)(xln2)=0-roots\ no\]

    There are no maximum or minimum points.

    The function decreases over the entire domain of definition;

    $y^("")=\frac(1)(x^2ln2)$;

    Convexity and concavity intervals:

    \[\frac(1)(x^2ln2) >0\]

    The function is concave over the entire domain of definition;

    $(\mathop(lim)_(x\to 0) y\ )=+\infty ,\ (\mathop(lim)_(x\to +\infty ) y\ )=-\infty ,\ $;

Figure 3

The logarithmic function is denoted

its y corresponding to the value of x is called the natural number of x. By definition, relation (1) is equivalent to

(e - ). Since e y > 0 for any real y, then the logarithmic function is defined only for x > 0. In a more general sense, the logarithmic function is called the function

where a > 0 (a ¹ 1) is arbitrary. However, in mathematical analysis, the InX function has a special feature; the function log a X is reduced to it by the formula:

where M = 1/In a. The logarithmic function is one of the main ones; her schedule rice. one) is called . The basic logarithmic functions follow from the corresponding properties of the exponential function and logarithms; e.g. The logarithmic function satisfies the functional equation

Rice. 1 to Art. Logarithmic function.

For 1< х, 1 справедливо разложение Логарифмическая функция в степенной ряд:

log(1 + x) = x

Many are expressed in terms of the logarithmic function; for example

,

.

The logarithmic function is constantly encountered in mathematical analysis and its applications.

The logarithmic function was well known in the 17th century. For the first time, the relationship between variables, expressed by the logarithmic function, was considered by J. (1614). He represented the relationship between numbers and their logarithms using two points moving along parallel straight lines ( rice. 2). One of them (Y) moves uniformly, starting from C, and the other (X), starting from A, moves with proportional to it to B. If we put SU = y, XB = x, then, according to this definition, dx / dy = kx, whence .

A logarithmic function on a complex one is a multi-valued (infinite-valued) function defined for all z ¹ 0 denoted by Lnz. An unambiguous branch of this function, defined as

Inz = In½ z½ + i arg z,

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The logarithmic function (80) performs an inverse mapping of the entire plane w with a cut into a strip - i / /: i, an infinite-sheeted Riemann surface onto a complete z - plane.


Logarithmic function: y logax, where the base of the logarithms is a-positive number, not equal to one.

The logarithmic function plays a special role in the development and analysis of algorithms, so it is worth considering in more detail. Because we often deal with analytical results where the constant factor is omitted, we use log TV notation, omitting the base. Changing the base of the logarithm changes the value of the logarithm only by a constant factor, however, special values ​​of the base of the logarithm arise in certain contexts.

The logarithmic function is the inverse of the exponential. Its graph (Fig. 247) is obtained from the graph of the exponential function (with the same base) by bending the drawing along the bisector of the first coordinate angle. The graph of any inverse function is also obtained.

The logarithmic function is then introduced as the reciprocal of the exponential. The properties of both functions are derived without difficulty from these definitions. It was this definition that received the approval of Gauss, who at the same time expressed disagreement with the assessment given to him in the review of the Göttingen Scientific News. At the same time, Gauss approached the issue from a broader point of view than da Cunha. The latter limited himself to considering the exponential and logarithmic functions in the real region, while Gauss extended their definition to complex variables.

The logarithmic function y logax is monotonic over the entire domain of its definition.

The logarithmic function is continuous and differentiable over the entire domain of definition.

The logarithmic function increases monotonically if a I, When 0 a 1, the logarithmic function with base a decreases monotonically.

The logarithmic function is defined only for positive values ​​of x and one-to-one displays the interval (0; 4 - oc.

The logarithmic function y loga x is the inverse function of the exponential function yax.

Logarithmic function: y ogax, where the base of the logarithms a is a positive number not equal to one.

Logarithmic functions are well combined with physical concepts of the nature of the creep of polyethylene under conditions where the strain rate is low. In this respect, they coincide with the Andraade equation, so they are sometimes used to approximate experimental data.

The logarithmic function, or natural logarithm, u In z, is determined by solving the transcendental equation r ei with respect to u. In the range of real values ​​of x and y, under the condition x 0, this equation admits a unique solution.

The section of logarithms is of great importance in the school course "Mathematical Analysis". Tasks for logarithmic functions are based on other principles than tasks for inequalities and equations. Knowledge of the definitions and basic properties of the concepts of logarithm and logarithmic function will ensure the successful solution of typical USE problems.

Before proceeding to explain what a logarithmic function is, it is worth referring to the definition of a logarithm.

Let's look at a specific example: a log a x = x, where a › 0, a ≠ 1.

The main properties of logarithms can be listed in several points:

Logarithm

Logarithm is a mathematical operation that allows using the properties of a concept to find the logarithm of a number or expression.

Examples:

Logarithm function and its properties

The logarithmic function has the form

We note right away that the graph of a function can be increasing for a › 1 and decreasing for 0 ‹ a ‹ 1. Depending on this, the function curve will have one form or another.

Here are the properties and method for plotting graphs of logarithms:

  • the domain of f(x) is the set of all positive numbers, i.e. x can take any value from the interval (0; + ∞);
  • ODZ functions - the set of all real numbers, i.e. y can be equal to any number from the interval (- ∞; +∞);
  • if the base of the logarithm a > 1, then f(x) increases over the entire domain of definition;
  • if the base of the logarithm is 0 ‹ a ‹ 1, then F is decreasing;
  • the logarithmic function is neither even nor odd;
  • the graph curve always passes through the point with coordinates (1;0).

Building both types of graphs is very simple, let's look at the process using an example

First you need to remember the properties of a simple logarithm and its function. With their help, you need to build a table for specific x and y values. Then, on the coordinate axis, the obtained points should be marked and connected by a smooth line. This curve will be the required graph.

The logarithmic function is the inverse of the exponential function given by y= a x . To verify this, it is enough to draw both curves on the same coordinate axis.

Obviously, both lines are mirror images of each other. By constructing a straight line y = x, you can see the axis of symmetry.

In order to quickly find the answer to the problem, you need to calculate the values ​​of the points for y = log 2⁡ x, and then simply move the origin of the coordinate points three divisions down the OY axis and 2 divisions to the left along the OX axis.

As proof, we will build a calculation table for the points of the graph y = log 2 ⁡ (x + 2) -3 and compare the obtained values ​​​​with the figure.

As you can see, the coordinates from the table and the points on the graph match, therefore, the transfer along the axes was carried out correctly.

Examples of solving typical USE problems

Most of the test tasks can be divided into two parts: finding the domain of definition, specifying the type of function according to the graph drawing, determining whether the function is increasing / decreasing.

For a quick answer to tasks, it is necessary to clearly understand that f (x) increases if the exponent of the logarithm a > 1, and decreases - when 0 ‹ a ‹ 1. However, not only the base, but also the argument can greatly affect the form of the function curve.

F(x) marked with a check mark are the correct answers. Doubts in this case are caused by examples 2 and 3. The “-” sign in front of log changes increasing to decreasing and vice versa.

Therefore, the graph y=-log 3⁡ x decreases over the entire domain of definition, and y= -log (1/3) ⁡x increases, despite the fact that the base is 0 ‹ a ‹ 1.

Answer: 3,4,5.

Answer: 4.

These types of tasks are considered easy and are estimated at 1-2 points.

Task 3.

Determine whether the function is decreasing or increasing and indicate the scope of its definition.

Y = log 0.7 ⁡(0.1x-5)

Since the base of the logarithm is less than one but greater than zero, the function of x is decreasing. According to the properties of the logarithm, the argument must also be greater than zero. Let's solve the inequality:

Answer: the domain of definition D(x) is the interval (50; + ∞).

Answer: 3, 1, OX axis, to the right.

Such tasks are classified as average and are estimated at 3-4 points.

Task 5. Find the range for a function:

It is known from the properties of the logarithm that the argument can only be positive. Therefore, we calculate the area of ​​​​admissible values ​​of the function. To do this, it will be necessary to solve a system of two inequalities.

The main properties of the logarithm, the graph of the logarithm, the domain of definition, the set of values, the basic formulas, the increase and decrease are given. Finding the derivative of the logarithm is considered. As well as integral, power series expansion and representation by means of complex numbers.

Definition of logarithm

Logarithm with base a is the y function (x) = log x, inverse to the exponential function with base a: x (y) = a y.

Decimal logarithm is the logarithm to the base of the number 10 : log x ≡ log 10 x.

natural logarithm is the logarithm to the base of e: ln x ≡ log e x.

2,718281828459045... ;
.

The graph of the logarithm is obtained from the graph of the exponential function by mirror reflection about the straight line y \u003d x. On the left are graphs of the function y (x) = log x for four values bases of the logarithm:a= 2 , a = 8 , a = 1/2 and a = 1/8 . The graph shows that for a > 1 the logarithm is monotonically increasing. As x increases, the growth slows down significantly. At 0 < a < 1 the logarithm is monotonically decreasing.

Properties of the logarithm

Domain, set of values, ascending, descending

The logarithm is a monotonic function, so it has no extremums. The main properties of the logarithm are presented in the table.

Domain 0 < x < + ∞ 0 < x < + ∞
Range of values - ∞ < y < + ∞ - ∞ < y < + ∞
Monotone increases monotonically decreases monotonically
Zeros, y= 0 x= 1 x= 1
Points of intersection with the y-axis, x = 0 No No
+ ∞ - ∞
- ∞ + ∞

Private values


The base 10 logarithm is called decimal logarithm and is marked like this:

base logarithm e called natural logarithm:

Basic logarithm formulas

Properties of the logarithm following from the definition of the inverse function:

The main property of logarithms and its consequences

Base replacement formula

Logarithm is the mathematical operation of taking the logarithm. When taking a logarithm, the products of factors are converted to sums of terms.

Potentiation is the mathematical operation inverse to logarithm. When potentiating, the given base is raised to the power of the expression on which the potentiation is performed. In this case, the sums of terms are converted into products of factors.

Proof of the basic formulas for logarithms

Formulas related to logarithms follow from formulas for exponential functions and from the definition of an inverse function.

Consider the property of the exponential function
.
Then
.
Apply the property of the exponential function
:
.

Let us prove the base change formula.
;
.
Setting c = b , we have:

Inverse function

The reciprocal of the base a logarithm is the exponential function with exponent a.

If , then

If , then

Derivative of the logarithm

Derivative of logarithm modulo x :
.
Derivative of the nth order:
.
Derivation of formulas > > >

To find the derivative of a logarithm, it must be reduced to the base e.
;
.

Integral

The integral of the logarithm is calculated by integrating by parts : .
So,

Expressions in terms of complex numbers

Consider the complex number function z:
.
Let's express a complex number z via module r and argument φ :
.
Then, using the properties of the logarithm, we have:
.
Or

However, the argument φ not clearly defined. If we put
, where n is an integer,
then it will be the same number for different n.

Therefore, the logarithm, as a function of a complex variable, is not a single-valued function.

Power series expansion

For , the expansion takes place:

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Higher Educational Institutions, Lan, 2009.

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