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" according to its base "a" is considered the power of "c", to which it is necessary to raise the base "a", 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:
- Natural logarithm ln a, where the base is the Euler number (e = 2.7).
- Decimal a, where the base is 10.
- 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, the task was given to find the answer to the equation 10 x \u003d 100. It is very easy, you need to choose such a power, 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.
- 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.
- 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.
- The logarithm of the quotient looks like this: log a (s 1 / s 2) = log a s 1 - log a s 2.
- 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.
- 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.
- 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.
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 xx
0
for each predetermined 
, there is a number 
that as soon as 
, That 
.
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 
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
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Exponential function
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logarithmic function
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trigonometric function
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Inverse trigonometric function
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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.
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:
- log a x+log a y= log a (x · y);
- 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. Please note: the key point here is - 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. Many tests are based on this fact. 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 solving 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. It's called the 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.
- 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.
- 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.
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 = b8 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 clog 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 c9 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.
logarithm positive number N to base(b> 0, b 1 ) is called exponent x , to which you need to raise b to get N .
Logarithm notation:
This entry is equivalent to the following:b x = N .
EXAMPLES: log 3 81 \u003d 4, since 3 4 \u003d 81;
Log 1/3 27 = – 3 , since (1/3) - 3 = 3 3 = 27 .
The above definition of the logarithm can be written as an identity:
Basic properties of logarithms.
1) log b= 1 , because b 1 = b.
b
2) log 1 = 0 , because b 0 = 1 .
b
3) The logarithm of the product is equal to the sum of the logarithms of the factors:
log( ab) = log a+log b.
4) The logarithm of the quotient is equal to the difference between the logarithms of the dividend and the divisor:
log( a/b) = log a–log b.
5) The logarithm of the degree is equal to the product of the exponent and the logarithm of its base:
log (b k ) = k log b.
The consequence of this property is the following:log root equals the logarithm of the root number divided by the power of the root:
6) If the base of the logarithm is a power, then the value the reciprocal of the exponent can be taken out of the sign of the log rhyme:
The last two properties can be combined into one:
7) Transition modulus formula (ie. e . transition from one baselogarithm to another base):
In a particular case, when N = a we have:
Decimal logarithm called base logarithm 10. It is designated lg , i.e. log 10 N = lg N. Logarithms of numbers 10, 100, 1000, ... p are respectively 1, 2, 3, …,those. have so many positive
units, how many zeros are in the logarithm number after one. Logarithms of numbers 0.1, 0.01, 0.001, ... p avny respectively –1, –2, –3, …, i.e. have as many negative ones as there are zeros in the logarithm number before the one ( counting and zero integers). Logarithms other numbers have a fractional part called mantissa. Wholepart of the logarithm is called characteristic. For practicaldecimal logarithms are most convenient.
natural logarithm
called base logarithm
e. It is denoted ln , i.e. log eN
=
ln N. Number eis irrational,approximate value is 2.718281828. It is the limit towards which the number tends(1 + 1
/
n)
n with unlimited increasen(cm. first wonderful limit ).
Strange as it may seem, natural logarithms turned out to be very convenient when carrying out various operations related to the analysis of functions.Calculating base logarithmsemuch faster than any other basis.
