The number e is the base of the natural logarithm. Understanding the natural logarithm

The logarithm of a number b to base a is the exponent to which the number a must be raised to obtain the number b.

If, then.

Logarithm - extreme important mathematical quantity , since logarithmic calculus allows not only to solve exponential equations, but also operate with indicators, differentiate exponential and logarithmic functions, integrate them and bring them to a more acceptable form to be calculated.

In contact with

All properties of logarithms are directly related to the properties exponential functions. For example, the fact that means that:

It should be noted that when solving specific problems, the properties of logarithms may turn out to be more important and useful than the rules for working with powers.

Let us present some identities:

Here are the basic algebraic expressions:

;

.

Attention! can exist only for x>0, x≠1, y>0.

Let's try to understand the question of what natural logarithms are. Special interest in mathematics represent two types- the first one has the number “10” as its base, and is called the “decimal logarithm”. The second one is called natural. Base natural logarithm- number "e". This is what we will talk about in detail in this article.

Designations:

• lg x - decimal;
• ln x - natural.

Using the identity, we can see that ln e = 1, as well as the fact that lg 10=1.

Natural logarithm graph

Let's construct a graph of the natural logarithm using the standard classical method point by point. If you wish, you can check whether we are constructing the function correctly by examining the function. However, it makes sense to learn how to build it “manually” in order to know how to correctly calculate the logarithm.

Function: y = ln x. Let's write down a table of points through which the graph will pass:

Let us explain why we chose these particular values ​​of the argument x. It's all about identity: . For the natural logarithm this identity will look like this:

For convenience, we can take five reference points:

;

;

.

;

.

Thus, calculating natural logarithms is a fairly simple task; moreover, it simplifies calculations of operations with powers, turning them into ordinary multiplication.

By plotting a graph point by point, we get an approximate graph:

The domain of definition of the natural logarithm (i.e., all valid values ​​of the argument X) is all numbers greater than zero.

Attention! The domain of definition of the natural logarithm includes only positive numbers! The scope of definition does not include x=0. This is impossible based on the conditions for the existence of the logarithm.

The range of values ​​(i.e. all valid values ​​of the function y = ln x) is all numbers in the interval.

Natural log limit

Studying the graph, the question arises - how does the function behave at y<0.

Obviously, the graph of the function tends to cross the y-axis, but will not be able to do this, since the natural logarithm of x<0 не существует.

Limit of natural log can be written this way:

Formula for replacing the base of a logarithm

Dealing with a natural logarithm is much easier than dealing with a logarithm that has an arbitrary base. That is why we will try to learn how to reduce any logarithm to a natural one, or express it to an arbitrary base through natural logarithms.

Let's start with the logarithmic identity:

Then any number or variable y can be represented as:

where x is any number (positive according to the properties of the logarithm).

This expression can be taken logarithmically on both sides. Let's do this using an arbitrary base z:

Let’s use the property (only instead of “c” we have the expression):

From here we get the universal formula:

.

In particular, if z=e, then:

.

We were able to represent a logarithm to an arbitrary base through the ratio of two natural logarithms.

We solve problems

In order to better understand natural logarithms, let's look at examples of several problems.

Problem 1. It is necessary to solve the equation ln x = 3.

Solution: Using the definition of the logarithm: if , then , we get:

Problem 2. Solve the equation (5 + 3 * ln (x - 3)) = 3.

Solution: Using the definition of the logarithm: if , then , we get:

.

Let's use the definition of a logarithm again:

.

Thus:

.

You can approximately calculate the answer, or you can leave it in this form.

Task 3. Solve the equation.

Solution: Let's make a substitution: t = ln x. Then the equation will take the following form:

.

We have a quadratic equation. Let's find its discriminant:

First root of the equation:

.

Second root of the equation:

.

Remembering that we made the substitution t = ln x, we get:

In statistics and probability theory, logarithmic quantities are found very often. This is not surprising, because the number e often reflects the growth rate of exponential quantities.

In computer science, programming and computer theory, logarithms are encountered quite often, for example, in order to store N bits in memory.

In the theories of fractals and dimensions, logarithms are constantly used, since the dimensions of fractals are determined only with their help.

In mechanics and physics There is no section where logarithms were not used. Barometric distribution, all the principles of statistical thermodynamics, the Tsiolkovsky equation, etc. are processes that can be mathematically described only using logarithms.

In chemistry, logarithms are used in Nernst equations and descriptions of redox processes.

Amazingly, even in music, in order to find out the number of parts of an octave, logarithms are used.

Natural logarithm Function y=ln x its properties

Proof of the main property of the natural logarithm

Lesson and presentation on the topics: "Natural logarithms. The base of the natural logarithm. The logarithm of a natural number"

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What is natural logarithm

Guys, in the last lesson we learned a new, special number - e. Today we will continue to work with this number.
We have studied logarithms and we know that the base of a logarithm can be many numbers that are greater than 0. Today we will also look at a logarithm whose base is the number e. Such a logarithm is usually called the natural logarithm. It has its own notation: $\ln(n)$ is the natural logarithm. This entry is equivalent to the entry: $\log_e(n)=\ln(n)$.
Exponential and logarithmic functions are inverses, then the natural logarithm is the inverse of the function: $y=e^x$.
Inverse functions are symmetric with respect to the straight line $y=x$.
Let's plot the natural logarithm by plotting the exponential function with respect to the straight line $y=x$.

It is worth noting that the angle of inclination of the tangent to the graph of the function $y=e^x$ at point (0;1) is 45°. Then the angle of inclination of the tangent to the graph of the natural logarithm at point (1;0) will also be equal to 45°. Both of these tangents will be parallel to the line $y=x$. Let's diagram the tangents:

Properties of the function $y=\ln(x)$

1. $D(f)=(0;+∞)$.
2. Is neither even nor odd.
3. Increases throughout the entire domain of definition.
4. Not limited from above, not limited from below.
5. There is no greatest value, no minimum value.
6. Continuous.
7. $E(f)=(-∞; +∞)$.
8. Convex upward.
9. Differentiable everywhere.

In the course of higher mathematics it is proven that the derivative of an inverse function is the inverse of the derivative of a given function.
There is not much point in going into the proof, let's just write the formula: $y"=(\ln(x))"=\frac(1)(x)$.

Example.
Calculate the value of the derivative of the function: $y=\ln(2x-7)$ at the point $x=4$.
Solution.
In general, our function is represented by the function $y=f(kx+m)$; we can calculate the derivatives of such functions.
$y"=(\ln((2x-7)))"=\frac(2)((2x-7))$.
Let's calculate the value of the derivative at the required point: $y"(4)=\frac(2)((2*4-7))=2$.
Answer: 2.

Example.
Draw a tangent to the graph of the function $y=ln(x)$ at the point $х=е$.
Solution.
We remember well the equation of the tangent to the graph of a function at the point $x=a$.
$y=f(a)+f"(a)(x-a)$.
We sequentially calculate the required values.
$a=e$.
$f(a)=f(e)=\ln(e)=1$.
$f"(a)=\frac(1)(a)=\frac(1)(e)$.
$y=1+\frac(1)(e)(x-e)=1+\frac(x)(e)-\frac(e)(e)=\frac(x)(e)$.
The tangent equation at the point $x=e$ is the function $y=\frac(x)(e)$.
Let's plot the natural logarithm and the tangent line.

Example.
Examine the function for monotonicity and extrema: $y=x^6-6*ln(x)$.
Solution.
The domain of definition of the function $D(y)=(0;+∞)$.
Let's find the derivative of the given function:
$y"=6*x^5-\frac(6)(x)$.
The derivative exists for all x from the domain of definition, then there are no critical points. Let's find stationary points:
$6*x^5-\frac(6)(x)=0$.
$\frac(6*x^6-6)(x)=0$.
$6*x^6-6=0$.
$x^6-1=0$.
$x^6=1$.
$x=±1$.
The point $х=-1$ does not belong to the domain of definition. Then we have one stationary point $x=1$. Let's find the intervals of increasing and decreasing:

Point $x=1$ is the minimum point, then $y_min=1-6*\ln(1)=1$.
Answer: The function decreases on the segment (0;1), the function increases on the ray $ (\displaystyle ). The simplicity of this definition, which is consistent with many other formulas that use this logarithm, explains the origin of the name "natural".

If we consider the natural logarithm as a real function of a real variable, then it is the inverse function of the exponential function, which leads to the identities:

e ln ⁡ a = a (a > 0) ; (\displaystyle e^(\ln a)=a\quad (a>0);) ln ⁡ e a = a (a > 0) . (\displaystyle \ln e^(a)=a\quad (a>0).)

Like all logarithms, the natural logarithm maps multiplication to addition:

ln ⁡ x y = ln ⁡ x + ln ⁡ y . (\displaystyle \ln xy=\ln x+\ln y.)