Logarithm of a number N based on A called exponent X , to which you need to build A to get the number N

Provided that
,
,

From the definition of logarithm it follows that
, i.e.
- this equality is fundamental logarithmic identity.

Logarithms to base 10 are called decimal logarithms. Instead of
write
.

Logarithms to the base e are called natural and are designated
.

Basic properties of logarithms.

The logarithm of one is equal to zero for any base.

The logarithm of the product is equal to the sum of the logarithms of the factors.

3) The logarithm of the quotient is equal to the difference of the logarithms

Factor
called the modulus of transition from logarithms to the base a to logarithms at the base b .

Using properties 2-5, it is often possible to reduce the logarithm of a complex expression to the result of simple arithmetic operations on logarithms.

For example,

Such transformations of a logarithm are called logarithms. Transformations inverse to logarithms are called potentiation.

Chapter 2. Elements of higher mathematics.

1. Limits

Limit of the function
is a finite number A if, as xx 0 for each predetermined
, there is such a number
that as soon as
, That
.

A function that has a limit differs from it by an infinitesimal amount:
, where- b.m.v., i.e.
.

Example. Consider the function
.

When striving
, function y tends to zero:

1.1. Basic theorems about limits.

The limit of a constant value is equal to this constant value

.

The limit of the sum (difference) of a finite number of functions is equal to the sum (difference) of the limits of these functions.

Limit of the product of a finite number of functions equal to the product the limits of these functions.

The limit of the quotient of two functions is equal to the quotient of the limits of these functions if the limit of the denominator is not zero.

Wonderful Limits

,
, Where

1.2. Limit Calculation Examples

However, not all limits are calculated so easily. More often, calculating the limit comes down to revealing an uncertainty of the type: or .

.

2. Derivative of a function

Let us have a function
, continuous on the segment
.

Argument got some increase
. Then the function will receive an increment
.

Argument value corresponds to the function value
.

Argument value
corresponds to the function value.

Hence, .

Let us find the limit of this ratio at
. If this limit exists, then it is called the derivative of the given function.

Definition 3 Derivative of a given function
by argument is called the limit of the ratio of the increment of a function to the increment of the argument, when the increment of the argument arbitrarily tends to zero.

Derivative of a function
can be designated as follows:

; ; ; .

Definition 4The operation of finding the derivative of a function is called differentiation.

2.1. Mechanical meaning of derivative.

Let us consider the rectilinear motion of some rigid body or material point.

Let at some point in time moving point
was at a distance from the starting position
.

After some period of time
she moved a distance
. Attitude =- average speed of a material point
. Let us find the limit of this ratio, taking into account that
.

Consequently, determining the instantaneous speed of motion of a material point is reduced to finding the derivative of the path with respect to time.

2.2. Geometric meaning derivative

Let us have a graphically defined function
.

Rice. 1. Geometric meaning of derivative

If
, then point
, will move along the curve, approaching the point
.

Hence
, i.e. the value of the derivative for a given value of the argument numerically equal to the tangent of the angle formed by the tangent at a given point with the positive direction of the axis
.

2.3. Table of basic differentiation formulas.

Power function

Exponential function

Logarithmic function

Trigonometric function

Inverse trigonometric function

2.4. Rules of differentiation.

Derivative of

Derivative of the sum (difference) of functions

Derivative of the product of two functions

Derivative of the quotient of two functions

2.5. Derivative of a complex function.

Let the function be given
such that it can be represented in the form

And
, where the variable is an intermediate argument, then

The derivative of a complex function is equal to the product of the derivative of the given function with respect to the intermediate argument and the derivative of the intermediate argument with respect to x.

Example 1.

Example 2.

3. Differential function.

Let there be
, differentiable on some interval
and let at this function has a derivative

,

then we can write

(1),

Where - an infinitesimal quantity,

since when

Multiplying all terms of equality (1) by
we have:

Where
- b.m.v. higher order.

Magnitude
called the differential of the function
and is designated

.

3.1. Geometric value of the differential.

Let the function be given
.

Fig.2. Geometric meaning of differential.

.

Obviously, the differential of the function
is equal to the increment of the ordinate of the tangent at a given point.

3.2. Derivatives and differentials of various orders.

If there is
, Then
is called the first derivative.

The derivative of the first derivative is called the second-order derivative and is written
.

Derivative of the nth order of the function
is called the (n-1)th order derivative and is written:

.

The differential of the differential of a function is called the second differential or second order differential.

.

.

3.3 Solving biological problems using differentiation.

Task 1. Studies have shown that the growth of a colony of microorganisms obeys the law
, Where N – number of microorganisms (in thousands), t – time (days).

b) Will the population of the colony increase or decrease during this period?

Answer. The size of the colony will increase.

Task 2. The water in the lake is periodically tested to monitor the content of pathogenic bacteria. Through t days after testing, the concentration of bacteria is determined by the ratio

.

When will the lake have a minimum concentration of bacteria and will it be possible to swim in it?

Solution: A function reaches max or min when its derivative is zero.

,

Let's determine the max or min will be in 6 days. To do this, let's take the second derivative.

Answer: After 6 days there will be a minimum concentration of bacteria.

What is a logarithm?

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

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 me? Fine. Now, in just 10 - 20 minutes you:

1. You will understand what is a logarithm.

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

3. Learn to calculate simple logarithms.

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

I feel like you have doubts... Well, okay, mark the time! Let's go!

First, solve this equation in your head:

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

Instructions

Write the given logarithmic expression. If the expression uses the logarithm of 10, then its notation is shortened and looks like this: lg b is the decimal logarithm. If the logarithm has the number e as its base, then write the expression: ln b – natural logarithm. It is understood that the result of any is the power to which the base number must be raised to obtain the number b.

When finding the sum of two functions, you simply need to differentiate them one by one and add the results: (u+v)" = u"+v";

When finding the derivative of the product of two functions, it is necessary to multiply the derivative of the first function by the second and add the derivative of the second function multiplied by the first function: (u*v)" = u"*v+v"*u;

In order to find the derivative of the quotient of two functions, it is necessary to subtract from the product of the derivative of the dividend multiplied by the divisor function the product of the derivative of the divisor multiplied by the function of the dividend, and divide all this by the divisor function squared. (u/v)" = (u"*v-v"*u)/v^2;

If given complex function, then it is necessary to multiply the derivative of internal function and the derivative of the external one. Let y=u(v(x)), then y"(x)=y"(u)*v"(x).

Using the results obtained above, you can differentiate almost any function. So let's look at a few examples:

y=x^4, y"=4*x^(4-1)=4*x^3;

y=2*x^3*(e^x-x^2+6), y"=2*(3*x^2*(e^x-x^2+6)+x^3*(e^x-2 *x));
There are also problems involving calculating the derivative at a point. Let the function y=e^(x^2+6x+5) be given, you need to find the value of the function at the point x=1.
1) Find the derivative of the function: y"=e^(x^2-6x+5)*(2*x +6).

2) Calculate the value of the function in given point y"(1)=8*e^0=8

Video on the topic

Useful advice

Learn the table of elementary derivatives. This will significantly save time.

Sources:

• derivative of a constant

So, what is the difference between rational equation from the rational? If the unknown variable is under the sign square root, then the equation is considered irrational.

Instructions

The main method for solving such equations is the method of constructing both sides equations into a square. However. this is natural, the first thing you need to do is get rid of the sign. This method is not technically difficult, but sometimes it can lead to trouble. For example, the equation is v(2x-5)=v(4x-7). By squaring both sides you get 2x-5=4x-7. Solving such an equation is not difficult; x=1. But the number 1 will not be given equations. Why? Substitute one into the equation instead of the value of x. And the right and left sides will contain expressions that do not make sense, that is. This value is not valid for a square root. Therefore, 1 is an extraneous root, and therefore this equation has no roots.

So, an irrational equation is solved using the method of squaring both its sides. And having solved the equation, it is necessary to cut off extraneous roots. To do this, substitute the found roots into the original equation.

Consider another one.
2х+vх-3=0
Of course, this equation can be solved using the same equation as the previous one. Move Compounds equations, which do not have a square root, in right side and then use the squaring method. solve the resulting rational equation and roots. But also another, more elegant one. Enter a new variable; vх=y. Accordingly, you will receive an equation of the form 2y2+y-3=0. That is, the usual quadratic equation. Find its roots; y1=1 and y2=-3/2. Next, solve two equations vх=1; vх=-3/2. The second equation has no roots; from the first we find that x=1. Don't forget to check the roots.

Solving identities is quite simple. To do this, it is necessary to carry out identical transformations until the set goal is achieved. Thus, with the help of simple arithmetic operations, the task at hand will be solved.

You will need

• - paper;
• - pen.

Instructions

The simplest of such transformations are algebraic abbreviated multiplications (such as the square of the sum (difference), difference of squares, sum (difference), cube of the sum (difference)). In addition, there are many and trigonometric formulas, which are essentially the same identities.

Indeed, the square of the sum of two terms is equal to the square of the first plus twice the product of the first by the second and plus the square of the second, that is, (a+b)^2= (a+b)(a+b)=a^2+ab +ba+b ^2=a^2+2ab+b^2.

Simplify both

General principles of the solution

Repeat from a textbook on mathematical analysis or higher mathematics what a definite integral is. As is known, the solution definite integral there is a function whose derivative gives an integrand. This function is called an antiderivative. Based on this principle, the main integrals are constructed.
Determine by the type of the integrand which of the table integrals is suitable in this case. It is not always possible to determine this immediately. Often, the tabular form becomes noticeable only after several transformations to simplify the integrand.

Variable Replacement Method

If the integrand is a trigonometric function whose argument is a polynomial, then try using the change of variables method. In order to do this, replace the polynomial in the argument of the integrand with some new variable. Based on the relationship between the new and old variables, determine the new limits of integration. Differentiation given expression find a new differential in . So you will get new look of the previous integral, close to or even corresponding to any tabular one.

Solving integrals of the second kind

If the integral is an integral of the second kind, a vector form of the integrand, then you will need to use the rules for the transition from these integrals to scalar ones. One such rule is the Ostrogradsky-Gauss relation. This law allows you to go from the rotor flux of some vector function to the triple integral over the divergence of a given vector field.

Substitution of integration limits

After finding the antiderivative, it is necessary to substitute the limits of integration. First, substitute the value of the upper limit into the expression for the antiderivative. You will get some number. Next, subtract from the resulting number another number obtained from the lower limit into the antiderivative. If one of the limits of integration is infinity, then when substituting it into antiderivative function it is necessary to go to the limit and find what the expression strives for.
If the integral is two-dimensional or three-dimensional, then you will have to represent the limits of integration geometrically to understand how to evaluate the integral. Indeed, in the case of, say, a three-dimensional integral, the limits of integration can be entire planes that limit the volume being integrated.

(from Greek λόγος - “word”, “relation” and ἀριθμός - “number”) numbers b based on a(log α b) is called such a number c, And b= a c, that is, records log α b=c And b=ac are equivalent. The logarithm makes sense if a > 0, a ≠ 1, b > 0.

In other words logarithm numbers b based on A formulated as an exponent to which a number must be raised a to get the number b(logarithm exists only for positive numbers).

From this formulation it follows that the calculation x= log α b, is equivalent to solving the equation a x =b.

For example:

log 2 8 = 3 because 8 = 2 3 .

Let us emphasize that the indicated formulation of the logarithm makes it possible to immediately determine logarithm value, when the number under the logarithm sign acts as a certain power of the base. Indeed, the formulation of the logarithm makes it possible to justify that if b=a c, then the logarithm of the number b based on a equals With. It is also clear that the topic of logarithms is closely related to the topic powers of a number.

Calculating the logarithm is called logarithm. Logarithm is the mathematical operation of taking a logarithm. When taking logarithms, products of factors are transformed into sums of terms.

Potentiation is the inverse mathematical operation of logarithm. During potentiation, a given base is raised to the degree of expression over which potentiation is performed. In this case, the sums of terms are transformed into a product of factors.

Quite often, real logarithms are used with bases 2 (binary), Euler's number e ≈ 2.718 (natural logarithm) and 10 (decimal).

At this stage it is advisable to consider logarithm samples log 7 2 , ln √ 5, lg0.0001.

And the entries lg(-3), log -3 3.2, log -1 -4.3 do not make sense, since in the first of them a negative number is placed under the logarithm sign, in the second - negative number in the base, and in the third - both a negative number under the logarithm sign and a unit in the base.

Conditions for determining the logarithm.

It is worth considering separately the conditions a > 0, a ≠ 1, b > 0.under which we get definition of logarithm. Let's look at why these restrictions were taken. An equality of the form x = log α will help us with this b, called the basic logarithmic identity, which directly follows from the definition of logarithm given above.

Let's take the condition a≠1. Since one to any power is equal to one, then the equality x=log α b can only exist when b=1, but log 1 1 will be any real number. To eliminate this ambiguity, we take a≠1.

Let us prove the necessity of the condition a>0. At a=0 according to the formulation of the logarithm can exist only when b=0. And accordingly then log 0 0 can be any non-zero real number, since zero to any non-zero power is zero. This ambiguity can be eliminated by the condition a≠0. And when a<0 we would have to reject the analysis of rational and irrational values ​​of the logarithm, since a degree with a rational and irrational exponent is defined only for non-negative bases. It is for this reason that the condition is stipulated a>0.

And the last condition b>0 follows from inequality a>0, since x=log α b, and the value of the degree with a positive base a always positive.

Features of logarithms.

Logarithms characterized by distinctive features, which led to their widespread use to significantly facilitate painstaking calculations. When moving “into the world of logarithms,” multiplication is transformed into a much easier addition, division is transformed into subtraction, and exponentiation and root extraction are transformed, respectively, into multiplication and division by the exponent.

Formulation of logarithms and table of their values ​​(for trigonometric functions) was first published in 1614 by the Scottish mathematician John Napier. Logarithmic tables, enlarged and detailed by other scientists, were widely used in scientific and engineering calculations, and remained relevant until they began to be used electronic calculators and computers.

Logarithmic expressions, solving examples. In this article we will look at problems related to solving logarithms. The tasks ask the question of finding the meaning of an expression. It should be noted that the concept of logarithm is used in many tasks and understanding its meaning is extremely important. As for the Unified State Exam, the logarithm is used when solving equations, in applied problems, and also in tasks related to the study of functions.

Let us give examples to understand the very meaning of the logarithm:

Basic logarithmic identity:

Properties of logarithms that must always be remembered:

*The logarithm of the product is equal to the sum of the logarithms of the factors.

* * *

*The logarithm of a quotient (fraction) is equal to the difference between the logarithms of the factors.

* * *

*The logarithm of an exponent is equal to the product of the exponent and the logarithm of its base.

* * *

*Transition to a new foundation

* * *

More properties:

* * *

The calculation of logarithms is closely related to the use of properties of exponents.

Let's list some of them:

The essence of this property lies in the fact that when transferring the numerator to the denominator and vice versa, the sign of the exponent changes to the opposite. For example:

A corollary from this property:

* * *

When raising a power to a power, the base remains the same, but the exponents are multiplied.

* * *

As you have seen, the concept of a logarithm itself is simple. The main thing is what is needed good practice, which gives a certain skill. Of course, knowledge of formulas is required. If the skill in converting elementary logarithms has not been developed, then when solving simple tasks you can easily make a mistake.

Practice, solve the simplest examples from the mathematics course first, then move on to more complex ones. In the future, I will definitely show how “ugly” logarithms are solved; these won’t appear on the Unified State Examination, but they are of interest, don’t miss them!

That's all! Good luck to you!

Sincerely, Alexander Krutitskikh

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