Difference of logarithms with the same base. Properties of logarithms and examples of their solutions

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 indicators. It was they who served for the further discovery of logarithms. Examples of using this function can be found almost everywhere where it is required to simplify cumbersome multiplication to simple addition. If you spend 10 minutes reading this article, we will explain to you what logarithms are and how to work with them. Simple and accessible language.

Definition in mathematics

The 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" by its base "a" is considered the power of "c", to which the base "a" must be raised, so that in the end 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 such a degree that from 2 to the required degree you get 8. Having done some calculations in your mind, we get the number 3! And rightly so, because 2 to the power of 3 gives the number 8 in the answer.

Varieties 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 distinct kinds of 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. The logarithm of any number b to the base a>1.

Each of them is solved in a standard way, including simplification, reduction and subsequent reduction to one logarithm using logarithmic theorems. To obtain the correct values ​​​​of logarithms, one should remember their properties and the order of actions in their decisions.

Rules and some restrictions

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

• the base "a" must always be greater than zero, and at the same time not be 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 be greater than zero.

How to solve logarithms?

For example, given the task to find the answer to the equation 10 x \u003d 100. It is very easy, you need to choose such a power by raising the number ten to which we get 100. This, of course, is 10 2 \u003d 100.

Now let's represent this expression as a logarithmic one. We get log 10 100 = 2. When solving logarithms, all actions practically converge to finding the degree to which the base of the logarithm must be entered in order to obtain a given number.

To accurately determine the value of an unknown degree, you must 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 mindset and knowledge of the multiplication table. However, larger values ​​will require a power table. It can be used even by those who do not understand anything at all in 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 in the cells, the values ​​of the numbers are determined, which 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 real 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 equation. For example, 3 4 =81 can be written as the logarithm of 81 to base 3, which is four (log 3 81 = 4). For negative powers, the rules are the same: 2 -5 = 1/32 we write 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 consider examples and solutions of equations a little lower, immediately after studying their properties. Now let's look at what inequalities look like and how to distinguish them from equations.

An expression of the following form is given: log 2 (x-1) > 3 - it is a logarithmic inequality, since the unknown value "x" is under the sign of the logarithm. And also in the expression two quantities are compared: the logarithm of the desired number in 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 of 2 x = √9) imply one or more specific numerical values ​​in the answer, while when solving the inequality, both the range of acceptable values ​​and the points breaking this function. As a consequence, the answer is not a simple set of individual numbers, as in the answer of the equation, but a continuous series or set of numbers.

Basic theorems about logarithms

When solving primitive tasks on 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 get acquainted with examples of equations later, let's first analyze each property in more detail.

1. The basic identity looks like this: a logaB =B. It only applies if 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 prerequisite is: d, s 1 and s 2 > 0; a≠1. You can give a proof for this formula of logarithms, with examples and a 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 get that s 1 *s 2 = a f1 *a f2 = a f1+f2 (degree properties ), and further by definition: log a (s 1 *s 2)= f 1 + f 2 = log a s1 + log a s 2, which was to be proved.
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 the following form: log a q b n = n/q log a b.

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

Let log a b \u003d t, it turns out a t \u003d b. If you raise both parts to the power m: a tn = b n ;

but since a tn = (a q) nt/q = b n , hence 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 logarithm problems are examples of equations and inequalities. They are found in almost all problem books, and are also included in the mandatory part of exams in mathematics. To enter a university or pass entrance tests in mathematics, you need to know how to solve such tasks correctly.

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

When solving logarithmic equations, it is necessary to determine what kind of logarithm we have before us: an example of an expression may contain a natural logarithm or a decimal one.

Here are examples ln100, ln1026. Their solution boils down to the fact that you need to determine the degree to which the base 10 will be equal to 100 and 1026, respectively. For solutions of natural logarithms, one must 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 main theorems on logarithms.

1. The property of the logarithm of the product can be used in tasks where it is necessary to decompose a large value of the number 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 degree of the logarithm, we managed to solve at first glance a complex and unsolvable expression. It is only necessary to factorize the base and then take the exponent values ​​out of the sign of the logarithm.

Tasks from the exam

Logarithms are often found in entrance exams, especially a lot of logarithmic problems in the Unified State Exam (state exam for all school graduates). Usually these tasks are present not only in part A (the easiest test part of the exam), but also in part C (the most difficult and voluminous tasks). The exam implies an accurate and perfect knowledge of the topic "Natural logarithms".

Examples and problem solving are taken from the official versions of the exam. 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.

• All logarithms are best reduced to the same base so that the solution is not cumbersome and confusing.
• All expressions under the sign of the logarithm are indicated as positive, therefore, when taking out the exponent of the exponent of the expression, which is under the sign of the logarithm and as its base, the expression remaining under the logarithm must be positive.

Today we will talk about logarithm formulas and give demonstration solution examples.

By themselves, they imply solution patterns according to the basic properties of logarithms. Before applying the logarithm formulas to the solution, we recall for you, first all the properties:

Now, based on these formulas (properties), we show examples of solving logarithms.

Examples of solving logarithms based on formulas.

Logarithm a positive number b in base a (denoted log a b) is the exponent to which a must be raised to get b, with b > 0, a > 0, and 1.

According to the definition log a b = x, which is equivalent to a x = b, so log a a x = x.

Logarithms, examples:

log 2 8 = 3, because 2 3 = 8

log 7 49 = 2 because 7 2 = 49

log 5 1/5 = -1, because 5 -1 = 1/5

Decimal logarithm is an ordinary logarithm, the base of which is 10. Denoted as lg.

log 10 100 = 2 because 10 2 = 100

natural logarithm- also the usual logarithm logarithm, but with the base e (e \u003d 2.71828 ... - an irrational number). Referred to as ln.

It is desirable to remember the formulas or properties of logarithms, because we will need them later when solving logarithms, logarithmic equations and inequalities. Let's work through each formula again with examples.

• Basic logarithmic identity
  a log a b = b

  8 2log 8 3 = (8 2log 8 3) 2 = 3 2 = 9

• The logarithm of the product is equal to the sum of the logarithms
  log a (bc) = log a b + log a c

  log 3 8.1 + log 3 10 = log 3 (8.1*10) = log 3 81 = 4

• The logarithm of the quotient is equal to the difference of the logarithms
  log a (b/c) = log a b - log a c

  9 log 5 50 /9 log 5 2 = 9 log 5 50- log 5 2 = 9 log 5 25 = 9 2 = 81

• Properties of the degree of a logarithmable number and the base of the logarithm

  The exponent of a logarithm number log a b m = mlog a b

  Exponent of the base of the logarithm log a n b =1/n*log a b

  log a n b m = m/n*log a b,

  if m = n, we get log a n b n = log a b

  log 4 9 = log 2 2 3 2 = log 2 3

• Transition to a new foundation
  log a b = log c b / log c a,

  if c = b, we get log b b = 1

  then log a b = 1/log b a

  log 0.8 3*log 3 1.25 = log 0.8 3*log 0.8 1.25/log 0.8 3 = log 0.8 1.25 = log 4/5 5/4 = -1

As you can see, the logarithm formulas are not as complicated as they seem. Now, having considered examples of solving logarithms, we can move on to logarithmic equations. We will consider examples of solving logarithmic equations in more detail in the article: "". Do not miss!

If you still have questions about the solution, write them in the comments to the article.

Note: decided to get an education of another class study abroad as an option.

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 example,

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 simply. 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 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 a
, 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
, 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. Through 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.

Let's start with properties of the logarithm of unity. Its formulation is as follows: the logarithm of unity is equal to zero, that is, log a 1=0 for any a>0 , a≠1 . The proof is straightforward: since a 0 =1 for any a that satisfies the above conditions a>0 and a≠1 , then the proven equality log a 1=0 immediately follows from the definition of the logarithm.

Let's give examples of application of the considered property: log 3 1=0 , lg1=0 and .

Let's move on to the next property: the logarithm of a number equal to the base is equal to one, i.e, log a a=1 for a>0 , a≠1 . Indeed, since a 1 =a for any a , then by the definition of the logarithm log a a=1 .

Examples of using this property of logarithms are log 5 5=1 , log 5.6 5.6 and lne=1 .

For example, log 2 2 7 =7 , log10 -4 =-4 and .

Logarithm of the product of two positive numbers x and y is equal to the product of the logarithms of these numbers: log a (x y)=log a x+log a y, a>0 , a≠1 . Let us prove the property of the logarithm of the product. Due to the properties of the degree a log a x+log a y =a log a x a log a y, and since by the main logarithmic identity a log a x =x and a log a y =y , then a log a x a log a y =x y . Thus, a log a x+log a y =x y , whence the required equality follows by the definition of the logarithm.

Let's show examples of using the property of the logarithm of the product: log 5 (2 3)=log 5 2+log 5 3 and .

The product logarithm property can be generalized to the product of a finite number n of positive numbers x 1 , x 2 , …, x n as log a (x 1 x 2 ... x n)= log a x 1 + log a x 2 +…+ log a x n . This equality is easily proved.

For example, the natural logarithm of a product can be replaced by the sum of three natural logarithms of the numbers 4 , e , and .

Logarithm of the quotient of two positive numbers x and y is equal to the difference between the logarithms of these numbers. The quotient logarithm property corresponds to a formula of the form , where a>0 , a≠1 , x and y are some positive numbers. The validity of this formula is proved like the formula for the logarithm of the product: since , then by the definition of the logarithm .

Here is an example of using this property of the logarithm: .

Let's move on to property of the logarithm of degree. The logarithm of a degree is equal to the product of the exponent and the logarithm of the modulus of the base of this degree. We write this property of the logarithm of the degree in the form of a formula: log a b p =p log a |b|, where a>0 , a≠1 , b and p are numbers such that the degree of b p makes sense and b p >0 .

We first prove this property for positive b . The basic logarithmic identity allows us to represent the number b as a log a b , then b p =(a log a b) p , and the resulting expression, due to the power property, is equal to a p log a b . So we arrive at the equality b p =a p log a b , from which, by the definition of the logarithm, we conclude that log a b p =p log a b .

It remains to prove this property for negative b . Here we note that the expression log a b p for negative b makes sense only for even exponents p (since the value of the degree b p must be greater than zero, otherwise the logarithm will not make sense), and in this case b p =|b| p . Then b p =|b| p =(a log a |b|) p =a p log a |b|, whence log a b p =p log a |b| .

For example, and ln(-3) 4 =4 ln|-3|=4 ln3 .

It follows from the previous property property of the logarithm from the root: the logarithm of the root of the nth degree is equal to the product of the fraction 1/n and the logarithm of the root expression, that is, , where a>0 , a≠1 , n is a natural number greater than one, b>0 .

The proof is based on the equality (see ), which is valid for any positive b , and the property of the logarithm of the degree: .

Here is an example of using this property: .

Now let's prove conversion formula to the new base of the logarithm kind . To do this, it suffices to prove the validity of the equality log c b=log a b log c a . The basic logarithmic identity allows us to represent the number b as a log a b , then log c b=log c a log a b . It remains to use the property of the logarithm of the degree: log c a log a b = log a b log c a. Thus, the equality log c b=log a b log c a is proved, which means that the formula for the transition to a new base of the logarithm is also proved.

Let's show a couple of examples of applying this property of logarithms: and .

The formula for moving to a new base allows you to move on to working with logarithms that have a “convenient” base. For example, it can be used to go to natural or decimal logarithms so that you can calculate the value of the logarithm from the table of logarithms. The formula for the transition to a new base of the logarithm also allows in some cases to find the value of a given logarithm, when the values ​​of some logarithms with other bases are known.

Often used is a special case of the formula for the transition to a new base of the logarithm for c=b of the form . This shows that log a b and log b a – . For example, .

Also often used is the formula , which is useful for finding logarithm values. To confirm our words, we will show how the value of the logarithm of the form is calculated using it. We have . To prove the formula it is enough to use the transition formula to the new base of the logarithm a: .

It remains to prove the comparison properties of logarithms.

Let us prove that for any positive numbers b 1 and b 2 , b 1 log a b 2 , and for a>1, the inequality log a b 1

Finally, it remains to prove the last of the listed properties of logarithms. We confine ourselves to proving its first part, that is, we prove that if a 1 >1 , a 2 >1 and a 1 1 is true log a 1 b>log a 2 b . The remaining statements of this property of logarithms are proved by a similar principle.

Let's use the opposite method. Suppose that for a 1 >1 , a 2 >1 and a 1 1 log a 1 b≤log a 2 b is true. By the properties of logarithms, these inequalities can be rewritten as and respectively, and from them it follows that log b a 1 ≤log b a 2 and log b a 1 ≥log b a 2, respectively. Then, by the properties of powers with the same bases, the equalities b log b a 1 ≥b log b a 2 and b log b a 1 ≥b log b a 2 must be satisfied, that is, a 1 ≥a 2 . Thus, we have arrived at a contradiction to the condition a 1

Bibliography.

• Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the Beginnings of Analysis: A Textbook for Grades 10-11 of General Educational Institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).

In relation to

the task of finding any of the three numbers from the other two, given, can be set. Given a and then N is found by exponentiation. If N are given and then a is found by extracting the root of the power x (or exponentiation). Now consider the case when, given a and N, it is required to find x.

Let the number N be positive: the number a is positive and not equal to one: .

Definition. The logarithm of the number N to the base a is the exponent to which you need to raise a to get the number N; the logarithm is denoted by

Thus, in equality (26.1), the exponent is found as the logarithm of N to the base a. Entries

have the same meaning. Equality (26.1) is sometimes called the basic identity of the theory of logarithms; in fact, it expresses the definition of the concept of the logarithm. By this definition, the base of the logarithm a is always positive and different from unity; the logarithmable number N is positive. Negative numbers and zero do not have logarithms. It can be proved that any number with a given base has a well-defined logarithm. Therefore equality entails . Note that the condition is essential here, otherwise the conclusion would not be justified, since the equality is true for any values ​​of x and y.

Example 1. Find

Decision. To get the number, you need to raise base 2 to the power Therefore.

You can record when solving such examples in the following form:

Example 2. Find .

Decision. We have

In examples 1 and 2, we easily found the desired logarithm by representing the logarithmable number as the degree of the base with a rational exponent. In the general case, for example, for etc., this cannot be done, since the logarithm has an irrational value. Let us pay attention to one question related to this statement. In § 12 we gave the concept of the possibility of determining any real power of a given positive number. This was necessary for the introduction of logarithms, which, in general, can be irrational numbers.

Consider some properties of logarithms.

Property 1. If the number and base are equal, then the logarithm is equal to one, and, conversely, if the logarithm is equal to one, then the number and base are equal.

Proof. Let By the definition of the logarithm, we have and whence

Conversely, let Then by definition

Property 2. The logarithm of unity to any base is equal to zero.

Proof. By the definition of the logarithm (the zero power of any positive base is equal to one, see (10.1)). From here

Q.E.D.

The converse statement is also true: if , then N = 1. Indeed, we have .

Before stating the following property of logarithms, we agree to say that two numbers a and b lie on the same side of a third number c if they are both either greater than c or less than c. If one of these numbers is greater than c and the other is less than c, then we say that they lie on opposite sides of c.

Property 3. If the number and base lie on the same side of unity, then the logarithm is positive; if the number and base lie on opposite sides of unity, then the logarithm is negative.

The proof of property 3 is based on the fact that the degree of a is greater than one if the base is greater than one and the exponent is positive, or the base is less than one and the exponent is negative. The degree is less than one if the base is greater than one and the exponent is negative, or the base is less than one and the exponent is positive.

There are four cases to be considered:

We confine ourselves to the analysis of the first of them, the reader will consider the rest on his own.

Let then the exponent in equality be neither negative nor equal to zero, therefore, it is positive, i.e., which was required to be proved.

Example 3. Find out which of the following logarithms are positive and which are negative:

Solution, a) since the number 15 and the base 12 are located on the same side of the unit;

b) , since 1000 and 2 are located on the same side of the unit; at the same time, it is not essential that the base is greater than the logarithmic number;

c), since 3.1 and 0.8 lie on opposite sides of unity;

G) ; why?

e) ; why?

The following properties 4-6 are often called the rules of logarithm: they allow, knowing the logarithms of some numbers, to find the logarithms of their product, quotient, degree of each of them.

Property 4 (the rule for the logarithm of the product). The logarithm of the product of several positive numbers in a given base is equal to the sum of the logarithms of these numbers in the same base.

Proof. Let positive numbers be given.

For the logarithm of their product, we write the equality (26.1) defining the logarithm:

From here we find

Comparing the exponents of the first and last expressions, we obtain the required equality:

Note that the condition is essential; the logarithm of the product of two negative numbers makes sense, but in this case we get

In general, if the product of several factors is positive, then its logarithm is equal to the sum of the logarithms of the modules of these factors.

Property 5 (quotient logarithm rule). The logarithm of a quotient of positive numbers is equal to the difference between the logarithms of the dividend and the divisor, taken in the same base. Proof. Consistently find

Q.E.D.

Property 6 (rule of the logarithm of the degree). The logarithm of the power of any positive number is equal to the logarithm of that number times the exponent.

Proof. We write again the main identity (26.1) for the number :

Q.E.D.

Consequence. The logarithm of the root of a positive number is equal to the logarithm of the root number divided by the exponent of the root:

We can prove the validity of this corollary by presenting how and using property 6.

Example 4. Logarithm to base a:

a) (it is assumed that all values ​​b, c, d, e are positive);

b) (it is assumed that ).

Solution, a) It is convenient to pass in this expression to fractional powers:

Based on equalities (26.5)-(26.7) we can now write:

We notice that simpler operations are performed on the logarithms of numbers than on the numbers themselves: when multiplying numbers, their logarithms are added, when divided, they are subtracted, etc.

That is why logarithms have been used in computational practice (see Sec. 29).

The action inverse to the logarithm is called potentiation, namely: potentiation is the action by which this number itself is found by the given logarithm of a number. In essence, potentiation is not any special action: it comes down to raising the base to a power (equal to the logarithm of the number). The term "potentiation" can be considered synonymous with the term "exponentiation".

When potentiating, it is necessary to use the rules that are inverse to the rules of logarithm: replace the sum of logarithms with the logarithm of the product, the difference of logarithms with the logarithm of the quotient, etc. In particular, if there is any factor in front of the sign of the logarithm, then during potentiation it must be transferred to the indicator degrees under the sign of the logarithm.

Example 5. Find N if it is known that

Decision. In connection with the potentiation rule just stated, the factors 2/3 and 1/3, which are in front of the signs of the logarithms on the right side of this equality, will be transferred to the exponents under the signs of these logarithms; we get

Now we replace the difference of logarithms with the logarithm of the quotient:

to obtain the last fraction in this chain of equalities, we freed the previous fraction from irrationality in the denominator (section 25).

Property 7. If the base is greater than one, then the larger number has a larger logarithm (and the smaller one has a smaller one), if the base is less than one, then the larger number has a smaller logarithm (and the smaller one has a larger one).

This property is also formulated as a rule for the logarithm of inequalities, both parts of which are positive:

When taking the logarithm of inequalities to a base greater than one, the inequality sign is preserved, and when taking a logarithm to a base less than one, the sign of the inequality is reversed (see also item 80).

The proof is based on properties 5 and 3. Consider the case when If , then and, taking the logarithm, we obtain

(a and N/M lie on the same side of unity). From here

Case a follows, the reader will figure it out for himself.

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