How to find the logarithm of a logarithm. Logarithmic equation: basic formulas and techniques

As you know, when multiplying expressions with powers, their exponents always add up (a b *a c = a b+c). This mathematical law was derived by Archimedes, and later, in the 8th century, the mathematician Virasen created a table of integer exponents. It was they who served for the further discovery of logarithms. Examples of using this function can be found almost everywhere where you need to simplify cumbersome multiplication by simple addition. If you spend 10 minutes reading this article, we will explain to you what logarithms are and how to work with them. In simple and accessible language.

Definition in mathematics

A logarithm is an expression of the following form: log a b=c, that is, the logarithm of any non-negative number (that is, any positive) “b” to its base “a” is considered to be the power “c” to which the base “a” must be raised in order to ultimately get the value "b". Let's analyze the logarithm using examples, let's say there is an expression log 2 8. How to find the answer? It’s very simple, you need to find a power such that from 2 to the required power you get 8. After doing some calculations in your head, we get the number 3! And that’s true, because 2 to the power of 3 gives the answer as 8.

Types of logarithms

For many pupils and students, this topic seems complicated and incomprehensible, but in fact logarithms are not so scary, the main thing is to understand their general meaning and remember their properties and some rules. There are three individual species logarithmic expressions:

1. Natural logarithm ln a, where the base is the Euler number (e = 2.7).
2. Decimal a, where the base is 10.
3. Logarithm of any number b to base a>1.

Each of them is decided in a standard way, which includes simplification, reduction and subsequent reduction to one logarithm using logarithmic theorems. To obtain the correct values ​​of logarithms, you should remember their properties and the sequence of actions when solving them.

Rules and some restrictions

In mathematics, there are several rules-constraints that are accepted as an axiom, that is, they are not subject to discussion and are the truth. For example, it is impossible to divide numbers by zero, and it is also impossible to extract the even root of negative numbers. Logarithms also have their own rules, following which you can easily learn to work even with long and capacious logarithmic expressions:

• The base “a” must always be greater than zero, and not equal to 1, otherwise the expression will lose its meaning, because “1” and “0” to any degree are always equal to their values;
• if a > 0, then a b >0, it turns out that “c” must also be greater than zero.

How to solve logarithms?

For example, the task is given to find the answer to the equation 10 x = 100. This is very easy, you need to choose a power by raising the number ten to which we get 100. This, of course, is 10 2 = 100.

Now let's represent this expression in logarithmic form. We get log 10 100 = 2. When solving logarithms, all actions practically converge to find the power to which it is necessary to enter the base of the logarithm in order to obtain a given number.

To accurately determine the value of an unknown degree, you need to learn how to work with a table of degrees. It looks like this:

As you can see, some exponents can be guessed intuitively if you have a technical mind and knowledge of the multiplication table. However for large values you will need a table of degrees. It can be used even by those who know nothing at all about complex mathematical topics. The left column contains numbers (base a), the top row of numbers is the value of the power c to which the number a is raised. At the intersection, the cells contain the number values ​​that are the answer (a c =b). Let's take, for example, the very first cell with the number 10 and square it, we get the value 100, which is indicated at the intersection of our two cells. Everything is so simple and easy that even the most true humanist will understand!

Equations and inequalities

It turns out that under certain conditions the exponent is the logarithm. Therefore, any mathematical numerical expressions can be written as a logarithmic equality. For example, 3 4 =81 can be written as the base 3 logarithm of 81 equal to four (log 3 81 = 4). For negative powers the rules are the same: 2 -5 = 1/32 we write it as a logarithm, we get log 2 (1/32) = -5. One of the most fascinating sections of mathematics is the topic of “logarithms”. We will look at examples and solutions of equations below, immediately after studying their properties. Now let's look at what inequalities look like and how to distinguish them from equations.

The following expression is given: log 2 (x-1) > 3 - it is a logarithmic inequality, since the unknown value “x” is under the logarithmic sign. And also in the expression two quantities are compared: the logarithm of the desired number to base two is greater than the number three.

The most important difference between logarithmic equations and inequalities is that equations with logarithms (for example, the logarithm 2 x = √9) imply one or more specific answers. numerical values, while when solving the inequalities are defined as the region acceptable values, and the breakpoints of this function. As a consequence, the answer is not a simple set of individual numbers, as in the answer to an equation, but a continuous series or set of numbers.

Basic theorems about logarithms

When solving primitive tasks of finding the values ​​of the logarithm, its properties may not be known. However, when it comes to logarithmic equations or inequalities, first of all, it is necessary to clearly understand and apply in practice all the basic properties of logarithms. We will look at examples of equations later; let's first look at each property in more detail.

1. The main identity looks like this: a logaB =B. It applies only when a is greater than 0, not equal to one, and B is greater than zero.
2. The logarithm of the product can be represented in the following formula: log d (s 1 * s 2) = log d s 1 + log d s 2. In this case, the mandatory condition is: d, s 1 and s 2 > 0; a≠1. You can give a proof for this logarithmic formula, with examples and solution. Let log a s 1 = f 1 and log a s 2 = f 2, then a f1 = s 1, a f2 = s 2. We obtain that s 1 * s 2 = a f1 *a f2 = a f1+f2 (properties of degrees ), and then by definition: log a (s 1 * s 2) = f 1 + f 2 = log a s1 + log a s 2, which is what needed to be proven.
3. The logarithm of the quotient looks like this: log a (s 1/ s 2) = log a s 1 - log a s 2.
4. The theorem in the form of a formula takes on next view: log a q b n = n/q log a b.

This formula is called the “property of the degree of logarithm.” It resembles the properties of ordinary degrees, and it is not surprising, because all mathematics is based on natural postulates. Let's look at the proof.

Let log a b = t, it turns out a t =b. If we raise both parts to the power m: a tn = b n ;

but since a tn = (a q) nt/q = b n, therefore log a q b n = (n*t)/t, then log a q b n = n/q log a b. The theorem has been proven.

Examples of problems and inequalities

The most common types of problems on logarithms are examples of equations and inequalities. They are found in almost all problem books, and are also a required part of mathematics exams. To enter a university or pass entrance examinations in mathematics, you need to know how to correctly solve such tasks.

Unfortunately, there is no single plan or scheme for solving and determining the unknown value of the logarithm, however, it can be applied to every mathematical inequality or logarithmic equation certain rules. First of all, you should find out whether the expression can be simplified or lead to general appearance. You can simplify long logarithmic expressions if you use their properties correctly. Let's get to know them quickly.

When solving logarithmic equations, we must determine what type of logarithm we have: an example expression may contain a natural logarithm or a decimal one.

Here are examples ln100, ln1026. Their solution boils down to the fact that they need to determine the power to which the base 10 will be equal to 100 and 1026, respectively. To solve natural logarithms, you need to apply logarithmic identities or their properties. Let's look at examples of solving logarithmic problems of various types.

How to Use Logarithm Formulas: With Examples and Solutions

So, let's look at examples of using the basic theorems about logarithms.

1. The property of the logarithm of a product can be used in tasks where it is necessary to expand great importance numbers b into simpler factors. For example, log 2 4 + log 2 128 = log 2 (4*128) = log 2 512. The answer is 9.
2. log 4 8 = log 2 2 2 3 = 3/2 log 2 2 = 1.5 - as you can see, using the fourth property of the logarithm power, we managed to solve a seemingly complex and unsolvable expression. You just need to factor the base and then take the exponent values ​​out of the sign of the logarithm.

Assignments from the Unified State Exam

Logarithms are often found in entrance exams, especially many logarithmic problems in the Unified State Exam (state exam for all school graduates). Typically, these tasks are present not only in part A (the easiest test part of the exam), but also in part C (the most complex and voluminous tasks). The exam requires accurate and perfect knowledge of the topic “Natural logarithms”.

Examples and solutions to problems are taken from official Unified State Exam options. Let's see how such tasks are solved.

Given log 2 (2x-1) = 4. Solution:
let's rewrite the expression, simplifying it a little log 2 (2x-1) = 2 2, by the definition of the logarithm we get that 2x-1 = 2 4, therefore 2x = 17; x = 8.5.

• It is best to reduce all logarithms to the same base so that the solution is not cumbersome and confusing.
• All expressions under the logarithm sign are indicated as positive, therefore, when the exponent of an expression that is under the logarithm sign and as its base is taken out as a multiplier, the expression remaining under the logarithm must be positive.

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

Definition

Natural logarithm is the function y = ln x, the inverse of the exponential, x = e y, and 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 natural logarithm (functions y = ln x) is obtained from the exponential graph mirror image relative to the straight line y = x.

The natural logarithm is defined for positive values ​​of the variable x. It increases monotonically in its domain of definition.

At 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 quite 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

ln 1 = 0

Basic formulas for natural logarithms

Formulas following 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 substitution formula:

Proofs of these formulas are presented in the section "Logarithm".

Inverse function

The inverse 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 modulus x:
.
Derivative of nth order:
.
Deriving formulas > > >

Integral

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

Expressions using complex numbers

Consider the function of the 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 you put
, where n is an integer,
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

When the expansion takes place:

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.

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 the 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

Helpful advice

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

Sources:

• derivative of a constant

So, what is the difference between an irrational equation and a rational one? If the unknown variable is under the square root sign, 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, to the 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 the simplest 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 according to the textbook mathematical analysis or higher mathematics, which is a definite integral. 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. By differentiating this expression, find the new differential in . So you will get the new kind 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 us to move from the rotor flux of a certain 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 lower limit into an antiderivative. If one of the limits of integration is infinity, then when substituting it into the antiderivative function, it is necessary to go to the limit and find what the expression tends to.
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 - a 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 consider 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 the use of electronic calculators and computers.

The basic properties of the logarithm, logarithm graph, domain of definition, set of values, basic formulas, increasing and decreasing are given. Finding the derivative of a logarithm is considered. As well as integral, power series expansion and representation using complex numbers.

Definition of logarithm

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

Decimal logarithm is the logarithm to the base of a 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 mirroring it with respect to the straight line y = x. On the left are graphs of the function y (x) = log a x for four values logarithm bases: a = 2 , a = 8 , a = 1/2 and a = 1/8 . The graph shows that when a > 1 the logarithm increases monotonically. As x increases, growth slows down significantly. At 0 < a < 1 the logarithm decreases monotonically.

Properties of the logarithm

Domain, set of values, increasing, decreasing

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

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

Private values

The base 10 logarithm is called decimal logarithm and is denoted as follows:

Logarithm to base e called natural logarithm:

Basic formulas for logarithms

Properties of the logarithm arising 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 a logarithm. When taking logarithms, products of factors are converted 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 products of factors.

Proof of 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
.
Let's apply the property of the exponential function
:
.

Let us prove the base replacement formula.
;
.
Assuming c = b, we have:

Inverse function

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

If , then

If , then

Derivative of logarithm

Derivative of the logarithm of modulus x:
.
Derivative of nth order:
.
Deriving 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 using 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 uniquely defined. If you 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

When the expansion takes place:

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.