Simple tasks with logarithms. Properties of logarithms and examples of their solutions

The main properties of the natural logarithm, graph, domain of definition, set of values, basic formulas, derivative, integral, expansion in a power series and representation of the function ln x by means of complex numbers are given.

Definition

natural logarithm is the function y = ln x, inverse to the exponent, x \u003d e y , and which is the logarithm to the base of the number e: ln x = log e x.

The natural logarithm is widely used in mathematics because its derivative has the simplest form: (ln x)′ = 1/ x.

Based definitions, the base of the natural logarithm is the number e:
e ≅ 2.718281828459045...;
.

Graph of the function y = ln x.

Graph of the natural logarithm (functions y = ln x) is obtained from the graph of the exponent by mirror reflection about the straight line y = x .

The natural logarithm is defined at positive values variable x . It monotonically increases on its domain of definition.

As x → 0 the limit of the natural logarithm is minus infinity ( - ∞ ).

As x → + ∞, the limit of the natural logarithm is plus infinity ( + ∞ ). For large x, the logarithm increases rather slowly. Any power function x a with a positive exponent a grows faster than the logarithm.

Properties of the natural logarithm

Domain of definition, set of values, extrema, increase, decrease

The natural logarithm is a monotonically increasing function, so it has no extrema. The main properties of the natural logarithm are presented in the table.

ln x values

log 1 = 0

Basic formulas for natural logarithms

Formulas arising from the definition of the inverse function:

The main property of logarithms and its consequences

Base replacement formula

Any logarithm can be expressed in terms of natural logarithms using the base change formula:

The proofs of these formulas are presented in the "Logarithm" section.

Inverse function

The reciprocal of the natural logarithm is the exponent.

If , then

If , then .

Derivative ln x

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

Integral

The integral is calculated by integration by parts:
.
So,

Expressions in terms of complex numbers

Consider a function of a complex variable z :
.
Let's express the complex variable z via module r and argument φ :
.
Using the properties of the logarithm, we have:
.
Or
.
The argument φ is not uniquely defined. If we put
, where n is an integer,
then it will be the same number for different n.

Therefore, the natural 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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(from the Greek λόγος - "word", "relation" and ἀριθμός - "number") numbers b by reason a(log α b) is called such a number c, and b= a c, that is, 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 by reason a formulated as an exponent to which a number must be raised a to get the number b(the 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 instance:

log 2 8 = 3 because 8=2 3 .

We note that the indicated formulation of the logarithm makes it possible to immediately determine logarithm value when the number under the sign of the logarithm is 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 by reason a equals With. It is also clear that the topic of logarithm is closely related to the topic degree of number.

The calculation of the logarithm is referred to logarithm. Logarithm is the mathematical operation of taking a logarithm. When taking a logarithm, the products of factors are transformed into 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 transformed into the product of factors.

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

At this stage, it is worth considering samples of logarithms 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 sign of the logarithm, in the second - a negative number in the base, and in the third - and a negative number under the sign of the logarithm and unit in the base.

Conditions for determining the logarithm.

It is worth considering separately the conditions a > 0, a ≠ 1, b > 0. definition of a logarithm. Let's consider why these restrictions are taken. This will help us with an equality of the form x = log α b, called the basic logarithmic identity, which directly follows from the definition of the logarithm given above.

Take the condition a≠1. Since one is equal to one to any power, 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 only exist when b=0. And then accordingly log 0 0 can be any non-zero real number, since zero to any non-zero power is zero. To eliminate this ambiguity, the condition a≠0. And when a<0 we would have to reject the analysis of rational and irrational values ​​of the logarithm, since the exponent with rational and irrational exponent is defined only for non-negative bases. It is for this reason that the condition a>0.

And the last condition b>0 follows from the inequality a>0, because 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 greatly facilitate painstaking calculations. In the transition "to the world of logarithms", multiplication is transformed into a much easier addition, division into subtraction, and raising to a power and taking a root are transformed into multiplication and division by an exponent, respectively.

The formulation of logarithms and a 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 electronic calculators and computers began to be used.

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 - no serious logarithmic problem can be solved without them. 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:

1. log a x+log a y= log a (x · y);
2. 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. Note: key moment here - same grounds. 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 "What is a logarithm"). Take a look at the examples and see:

log 6 4 + log 6 9.

Since the bases of logarithms are the same, we use the sum formula:
log 6 4 + log 6 9 = log 6 (4 9) = log 6 36 = 2.

Task. Find the value of the expression: log 2 48 − log 2 3.

The bases are the same, we use the difference formula:
log 2 48 - log 2 3 = log 2 (48: 3) = log 2 16 = 4.

Task. Find the value of the expression: log 3 135 − log 3 5.

Again, the bases are the same, so we have:
log 3 135 − log 3 5 = log 3 (135: 5) = log 3 27 = 3.

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, 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. And one more thing: learn to apply all formulas not only from left to right, but also vice versa, i.e. you can enter the numbers before the sign of the logarithm into the logarithm itself. This is what is most often required.

Task. 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:
log 7 49 6 = 6 log 7 49 = 6 2 = 12

Task. Find the value of the expression:

[Figure caption]

Note that the denominator is a logarithm whose base and argument are exact powers: 16 = 2 4 ; 49 = 72. We have:

[Figure caption]

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:

Let the logarithm log a x. Then for any number c such that c> 0 and c≠ 1, the equality is true:

[Figure caption]

In particular, if we put c = x, we get:

[Figure caption]

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:

Task. 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:

[Figure caption]

Since the product does not change from permutation of factors, we calmly multiplied four and two, and then figured out the logarithms.

Task. 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:

[Figure caption]

Now let's get rid of the decimal logarithm by moving to a new base:

[Figure caption]

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. That's what it's called: basic logarithmic identity.

Indeed, what will happen if the number b raise to the power so that b to this extent gives a 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.

Task. Find the value of the expression:

[Figure caption]

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:

[Figure caption]

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.

1. log a a= 1 is the logarithmic unit. Remember once and for all: the logarithm to any base a from this base itself is equal to one.
2. 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! Download the cheat sheet at the beginning of the lesson, print it out and solve the problems.

The logarithm of a number N by reason a is called exponent X , to which you need to raise a to get the number N

Provided that
,
,

It follows from the definition of the logarithm that
, i.e.
- this equality is the basic logarithmic identity.

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

base logarithms e are called natural and denoted
.

Basic properties of logarithms.

The logarithm of unity for any base is zero

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
is called the modulus of transition from logarithms at 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 instance,

Such transformations of the logarithm are called logarithms. Transformations reciprocal of logarithms are called potentiation.

Chapter 2. Elements of higher mathematics.

1. Limits

function limit
is a finite number A if, when striving xx 0 for each predetermined
, there is a number
that as soon as
, then
.

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

Example. Consider the function
.

When striving
, function y goes 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.

The limit of a product of a finite number of functions is equal to the product of 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 equal to zero.

Remarkable Limits

,
, where

1.2. Limit Calculation Examples

However, not all limits are calculated so easily. More often, the calculation of the limit is reduced to the disclosure of type uncertainty: or .

.

2. Derivative of a function

Let we have a function
, continuous on the segment
.

Argument got some boost
. Then the function will be incremented
.

Argument value corresponds to the value of the function
.

Argument value
corresponds to the value of the function .

Hence, .

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

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

Function derivative
can be denoted as follows:

; ; ; .

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

2.1. The mechanical meaning of the derivative.

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, the determination of the instantaneous velocity of a material point is reduced to finding the derivative of the path with respect to time.

2.2. Geometric value of the derivative

Suppose we have a graphically defined some function
.

Rice. 1. The geometric meaning of the derivative

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

Hence
, i.e. the value of the derivative given the value of the argument numerically equals 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. Differentiation rules.

Derivative of

Derivative of the sum (difference) of functions

Derivative of the product of two functions

The derivative of the quotient of two functions

2.5. Derivative of a complex function.

Let the function
such that it can be represented as

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 by the derivative of the intermediate argument with respect to x.

Example1.

Example2.

3. Function differential.

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

,

then you can write

(1),

where - an infinitesimal quantity,

because at

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

Where
- b.m.v. higher order.

Value
is called the differential of the function
and denoted

.

3.1. The geometric value of the differential.

Let the function
.

Fig.2. The geometric meaning of the differential.

.

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

3.2. Derivatives and differentials of various orders.

If there's
, 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 derivative of the (n-1) order and is written:

.

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

.

.

3.3 Solving biological problems using differentiation.

Task1. 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 colony will grow in size.

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

.

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

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

,

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

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