The natural logarithm of 0 is equal to. Logarithm

What is a logarithm?

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What is a logarithm? How to solve logarithms? These questions confuse many graduates. Traditionally, the topic of logarithms is considered complex, incomprehensible and scary. Especially - equations with logarithms.

This is absolutely not true. Absolutely! Don't believe? Fine. Now, for some 10 - 20 minutes you:

1. Understand what is a logarithm.

2. Learn to solve a whole class of exponential equations. Even if you haven't heard of them.

3. Learn to calculate simple logarithms.

Moreover, for this you will only need to know the multiplication table, and how a number is raised to a power ...

I feel you doubt ... Well, keep time! Go!

First, solve the following equation in your mind:

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You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

The logarithm of the number b to the base a is the exponent to which you need to raise the number a to get the number b.

If , then .

The logarithm is extremely important mathematical quantity, since the logarithmic calculus allows not only to solve exponential equations, but also operate with indicators, differentiate exponential and logarithmic functions, integrate them and lead to a more acceptable form to be calculated.

In contact with

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

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

Here are some identities:

Here are the main algebraic expressions:

;

.

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

Let's try to understand the question of what natural logarithms are. Separate interest in mathematics represent two types- the first has the number "10" at the base, and is called " decimal logarithm". The second is called natural. The base of the natural logarithm is the number e. It is about him that we will talk 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 that lg 10=1.

natural log graph

We construct a graph of the natural logarithm in the standard classical way by points. If you wish, you can check whether we are building a 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 = log x. Let's write a table of points through which the graph will pass:

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

For convenience, we can take five reference points:

;

;

.

;

.

Thus, counting natural logarithms is a fairly simple task, moreover, it simplifies the calculation of operations with powers, turning them into normal multiplication.

Having built a graph by points, we get an approximate graph:

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

Attention! The domain of definition of the natural logarithm includes only positive numbers! The scope 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 when 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.

Natural limit log can be written like this:

Formula for changing 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 in 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 logarithmized on both sides. Let's do this with an arbitrary base z:

Let's use the property (only instead of "with" we have an expression):

From here we get the universal formula:

.

In particular, if z=e, then:

.

We managed to represent the logarithm to an arbitrary base through the ratio of two natural logarithms.

We solve problems

In order to better navigate in natural logarithms, consider examples of several problems.

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

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

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

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

.

Once again, we apply the definition of the logarithm:

.

Thus:

.

You can calculate the answer approximately, 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 very common. This is not surprising, because the number e - often reflects the growth rate of exponential values.

In computer science, programming and computer theory, logarithms are quite common, 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. The barometric distribution, all the principles of statistical thermodynamics, the Tsiolkovsky equation and so on are processes that can only be described mathematically using logarithms.

In chemistry, the logarithm is used in the Nernst equations, 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

often take a number e = 2,718281828 . Logarithms in this base are called natural. When performing calculations with natural logarithms, it is common to operate with the sign ln, but not log; while the number 2,718281828 , defining the base, do not indicate.

In other words, the wording will look like: natural logarithm numbers X is the exponent to which the number is to be raised e, To obtain x.

So, ln(7,389...)= 2 because e 2 =7,389... . The natural logarithm of the number itself e= 1 because e 1 =e, and the natural logarithm of unity is equal to zero, since e 0 = 1.

The number itself e defines the limit of a monotone bounded sequence

calculated that e = 2,7182818284... .

Quite often, in order to fix a number in memory, the digits of the required number are associated with some outstanding date. The speed of remembering the first nine digits of a number e after the decimal point will increase if you note that 1828 is the year of Leo Tolstoy's birth!

To date, there are fairly complete tables of natural logarithms.

natural log graph(functions y=ln x) is a consequence of the plot of the exponent as a mirror image with respect to the straight line y = x and looks like:

The natural logarithm can be found for every positive real number a as the area under the curve y = 1/x from 1 before a.

The elementary nature of this formulation, which fits in with many other formulas in which the natural logarithm is involved, was the reason for the formation of the name "natural".

If we analyze natural logarithm, as a real function of a real variable, then it acts inverse function to an exponential function, which reduces to the identities:

ln(a)=a (a>0)

ln(e a)=a

By analogy with all logarithms, the natural logarithm converts multiplication to addition, division to subtraction:

ln(xy) = ln(x) + ln(y)

ln(x/y)= lnx - lny

The logarithm can be found for every positive base that is not equal to one, not just for e, but logarithms for other bases differ from the natural logarithm only by a constant factor, and are usually defined in terms of the natural logarithm.

Having analyzed natural log graph, we get that it exists for positive values ​​of the variable x. It monotonically increases on its domain of definition.

At x → 0 the limit of the natural logarithm is minus infinity ( -∞ ).At x → +∞ the limit of the natural logarithm is plus infinity ( + ∞ ). At large x the logarithm increases rather slowly. Any power function x a with a positive exponent a increases faster than the logarithm. The natural logarithm is a monotonically increasing function, so it has no extrema.

Usage natural logarithms very rational in the passage of higher mathematics. Thus, the use of the logarithm is convenient for finding the answer to equations in which the unknowns appear as an exponent. The use of natural logarithms in calculations makes it possible to greatly facilitate a large number of mathematical formulas. base logarithms e are present in solving a significant number of physical problems and are naturally included in the mathematical description of individual chemical, biological and other processes. Thus, logarithms are used to calculate the decay constant for a known half-life, or to calculate the decay time in solving problems of radioactivity. They play a leading role in many sections of mathematics and practical sciences, they are resorted to in the field of finance to solve a large number of problems, including in the calculation of compound interest.

Lesson and presentation on the topics: "Natural logarithms. Base of a natural logarithm. 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 the logarithm can be a set of numbers that are greater than 0. Today we will also consider the logarithm, which is based on the number e. Such a logarithm is usually called the natural logarithm. It has its own notation: $\ln(n)$ is the natural logarithm. This notation is equivalent to: $\log_e(n)=\ln(n)$.
The exponential and logarithmic functions are inverse, 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 slope of the tangent to the graph of the function $y=e^x$ at the point (0;1) is 45°. Then the slope of the tangent to the graph of the natural logarithm at the point (1; 0) will also be equal to 45°. Both of these tangents will be parallel to the line $y=x$. Let's sketch the tangents:

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

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

In the course of higher mathematics it is proved that the derivative of an inverse function is the reciprocal of the derivative of the given function.
It doesn't make much sense to delve 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 $x=e$.
Solution.
The equation of the tangent to the graph of the function, at the point $x=a$, we remember well.
$y=f(a)+f"(a)(x-a)$.
Let us 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.

Example.
Investigate the function for monotonicity and extrema: $y=x^6-6*ln(x)$.
Solution.
Domain of the function $D(y)=(0;+∞)$.
Find the derivative of the given function:
$y"=6*x^5-\frac(6)(x)$.
The derivative exists for all x from the domain, 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 $х=1$. Find the intervals of increase and decrease:

The point $x=1$ is the minimum point, then $y_min=1-6*\ln(1)=1$.
Answer: The function is decreasing on the segment (0;1], the function is increasing on the ray $)