Logarithms
History of logarithms
The name was introduced by Napier, comes from the Greek words logoz and ariumoz - it literally means “numbers of relations”. Logarithms were invented by Napier. Napier invented logarithms no later than 1594. Logarithms with base a introduced by the mathematics teacher Speidel. The word base is borrowed from the theory of powers and transferred to the theory of logarithms by Euler. The verb "logarithm" appeared in the 19th century with Koppe. Cauchy was the first to propose introducing different signs for decimal and natural logarithms. Notation close to modern was introduced by the German mathematician Pringsheim in 1893. It was he who denoted the logarithm natural number through ln. The definition of the logarithm as an indicator of the degree of a given base can be found in Wallis (1665), Bernoulli (1694).
Definition of logarithm
logarithm number b>0 to the base a>0, a ≠ 1 , is called the exponent to which the number a must be raised to get the number b.
The logarithm of the number b to the base a is denoted: log a b
Basic logarithmic identity
This equality is just another form of defining the logarithm. He is often called basic logarithmic identity.
Example
1. 3=log 2 8 because 2³=8
2. ½=log 3 √3 since 3= √3
3.3 log 3 1/5 =1/5
4. 2=log √5 5 because (√5)²=5
natural and decimal logarithms
natural is the logarithm whose base is e. Denoted ln b, i.e.
decimal the logarithm is called, the base of which is equal to 10. It is denoted lg b, i.e.
Basic properties of logarithms
Let: a > 0, a ≠ 1. Then:
1. log a x*y=logax+logay (x>0, y>0)
2. log a y/x=logax-logay (x>0, y>0)
3. log a x p =p*logax (x>0)
4. log a p x=1/p*logax (x>0)
Example
1) log 8 16+log 8 4= log 8 (16 4)= log 8 64= 2;
2) log 5 375– log 5 3= log 5 375/3=log 5 125= 3;
3) ½log 3 36+ log 3 2- log 3 √6- ½ log 3 8=log 3 √36+ log 3 2-(log 3 √6+log 3 √8) =log 3 12/4 √3=log 3 √3= ½.
Forms of conversion from a logarithm in one base to logarithms in another base
1. log a b=log c b/log c a
2.log a b=1/log b a
Logarithmic Equations
1) An equation containing a variable under the sign of the logarithm (log) is called logarithmic. The simplest example of a logarithmic equation is an equation of the form: log a x=b, where a>0 and a=1.
2) The solution of the logarithmic equation of the form: log a f(x)=log a g(x) (1) is based on the fact that it is equivalent to an equation of the form f(x) = g(x) (2) with additional conditions f(x)>0 and g(x)>0.
3) When passing from equation (1) to equation (2), extraneous roots may appear, therefore, verification is required for their identification.
4) When solving logarithmic equations, the substitution method is often used.
Conclusion
Logarithm a number that simplifies many complex arithmetic operations. Using their logarithms instead of numbers in calculations makes it possible to replace multiplication with a simpler operation of addition, division with subtraction, raising to a power with multiplication, and extracting roots with division.
The only way to implement long-distance travel there was navigation, which is always associated with the performance of large volumes of navigational calculations. Now it is difficult to imagine the process of exhausting calculations when multiplying and dividing five-six-digit numbers "manually". the theologian, by the nature of his main activity, doing trigonometric calculations at his leisure, guessed to replace the laborious procedure of multiplication with simple addition. He himself said that his goal was "to get rid of the difficulty and boredom of calculations that scare many away from the study of mathematics." Efforts were crowned with success - a mathematical apparatus was created, called the system of logarithms.
So what is a logarithm? The basis of logarithmic calculations is a different representation of the number: instead of the usual positional system, as we are used to, the number A is represented as power expression, where some arbitrary number N, called the base of the power, is raised to such a power of n that the result is the number A. Thus, n is the logarithm of the number A to the base N. The choice of the base of the logarithms determines the name of the system. For simple calculations, the decimal system of logarithms is used, and in science and technology, the system of natural logarithms is widely used, where the base is the irrational number e = 2.718. The expression that determines the logarithm of the number A, in the language of mathematics, is written as follows:
n=log(N)A, where N is the base of the degree.
Decimal and natural logarithms have their own specific abbreviations - lgA and lnA, respectively.
In a calculation system that uses the calculation of logarithms, the main element is the conversion of a number to a power form using a table of logarithms to some base, for example 10. This manipulation does not present any difficulties. Further, the property of power numbers is used, which consists in the fact that when multiplied, their powers are added. In practice, this means that the multiplication of numbers with a logarithmic representation is replaced by the addition of their powers. Therefore, the question “what is a logarithm”, if it is continued to “why do we need it”, has a simple answer - to simplify the procedure for multiplying and dividing multi-digit numbers - after all, addition “in a column” is much simpler than multiplication “in a column”. Who does not believe - let him try to add and multiply two eight-digit numbers.
The first tables of logarithms (based on c were published in 1614 by John Napier, and a completely error-free version, including tables of decimal logarithms, appeared in 1857 and is known as Bremiker's tables. The use of logarithms with a base in the form is due to the fact that the number e is quite simple get through the taylor series having wide application in the integral and
The essence of this computing system is contained in the answer to the question “what is a logarithm” and follows from the main logarithmic identity: N(base of the logarithm) n, equal to the logarithm of the number A(logA), is equal to this number A. In this case, A>0, i.e. . the logarithm is defined only for positive numbers, and the base of the logarithm is always greater than 0 and not equal to 1. Based on what has been said, the properties of the natural logarithm can be formulated as follows:
- The domain of the natural logarithm is the entire numerical axis from 0 to infinity.
- ln x \u003d 0 - a consequence of the well-known relation - any number to the zero power is equal to 1.
- ln (X*Y) = ln X + lnY - the most important property for computational manipulations - the logarithm of the product of two numbers is the sum of the logarithms of each of them.
- ln (X/Y) = ln X - lnY - the logarithm of the quotient of two numbers is equal to the difference of the logarithms of these numbers.
- ln (X)n =n*ln X .
- The natural logarithm is a differentiable, upward convex function, with ln' X = 1 / X
- log (N)A \u003d K * ln A - the logarithm for any positive base other than a number e differs from the natural one only by a coefficient.
Now every schoolchild knows what a logarithm is, but thanks to progress in the field of applied computer science computational problems are a thing of the past. However, logarithms, already as a mathematical tool, are used in solving equations with unknowns in the exponent, in expressions for finding time
What is a logarithm?
Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")
What is a logarithm? How to solve logarithms? These questions confuse many graduates. Traditionally, the topic of logarithms is considered complex, incomprehensible and scary. Especially - equations with logarithms.
This is absolutely not true. Absolutely! Don't believe? Good. Now, for some 10 - 20 minutes you:
1. Understand what is a logarithm.
2. Learn to solve a whole class exponential equations. Even if you haven't heard of them.
3. Learn to calculate simple logarithms.
Moreover, for this you will only need to know the multiplication table, and how a number is raised to a power ...
I feel you doubt ... Well, keep time! Go!
First, solve the following equation in your mind:
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)
you can get acquainted with functions and derivatives.
So, we have powers of two. If you take the number from the bottom line, then you can easily find the power to which you have to raise a two to get this number. For example, to get 16, you need to raise two to the fourth power. And to get 64, you need to raise two to the sixth power. This can be seen from the table.
And now - in fact, the definition of the logarithm:
The base a logarithm of the argument x is the power to which the number a must be raised to get the number x .
Notation: log a x \u003d b, where a is the base, x is the argument, b is actually what the logarithm is equal to.
For example, 2 3 = 8 ⇒ log 2 8 = 3 (the base 2 logarithm of 8 is three because 2 3 = 8). Might as well log 2 64 = 6, since 2 6 = 64.
The operation of finding the logarithm of a number to a given base is called the logarithm. So let's add a new row to our table:
| 2 1 | 2 2 | 2 3 | 2 4 | 2 5 | 2 6 |
| 2 | 4 | 8 | 16 | 32 | 64 |
| log 2 2 = 1 | log 2 4 = 2 | log 2 8 = 3 | log 2 16 = 4 | log 2 32 = 5 | log 2 64 = 6 |
Unfortunately, not all logarithms are considered so easily. For example, try to find log 2 5. The number 5 is not in the table, but logic dictates that the logarithm will lie somewhere on the segment. Because 2 2< 5 < 2 3 , а чем больше степень двойки, тем больше получится число.
Such numbers are called irrational: the numbers after the decimal point can be written indefinitely, and they never repeat. If the logarithm turns out to be irrational, it is better to leave it like this: log 2 5, log 3 8, log 5 100.
It is important to understand that the logarithm is an expression with two variables (base and argument). At first, many people confuse where the basis is and where the argument is. To avoid unfortunate misunderstandings just take a look at the picture:
Before us is nothing more than the definition of the logarithm. Remember: the logarithm is the power, to which you need to raise the base to get the argument. It is the base that is raised to a power - in the picture it is highlighted in red. It turns out that the base is always at the bottom! I tell this wonderful rule to my students at the very first lesson - and there is no confusion.
We figured out the definition - it remains to learn how to count logarithms, i.e. get rid of the "log" sign. To begin with, we note that two important facts follow from the definition:
- The argument and base must always be greater than zero. This follows from the definition of the degree rational indicator, to which the definition of the logarithm is reduced.
- The base must be different from unity, since a unit to any power is still a unit. Because of this, the question “to what power must one be raised to get two” is meaningless. There is no such degree!
Such restrictions are called valid range(ODZ). It turns out that the ODZ of the logarithm looks like this: log a x = b ⇒ x > 0, a > 0, a ≠ 1.
Note that there are no restrictions on the number b (the value of the logarithm) is not imposed. For example, the logarithm may well be negative: log 2 0.5 = −1, because 0.5 = 2 −1 .
However, for now we are only considering numeric expressions, where it is not required to know the ODZ of the logarithm. All restrictions have already been taken into account by the compilers of the problems. But when they go logarithmic equations and inequalities, DHS requirements will become mandatory. Indeed, in the basis and argument there can be very strong constructions, which do not necessarily correspond to the above restrictions.
Now consider the general scheme for calculating logarithms. It consists of three steps:
- Express the base a and the argument x as a power with the smallest possible base greater than one. Along the way, it is better to get rid of decimal fractions;
- Solve the equation for the variable b: x = a b ;
- The resulting number b will be the answer.
That's all! If the logarithm turns out to be irrational, this will be seen already at the first step. The requirement that the base be more than one, is very relevant: it reduces the probability of error and greatly simplifies calculations. Similar to decimals: if you immediately translate them into ordinary ones, there will be many times less errors.
Let's see how this scheme works with specific examples:
Task. Calculate the logarithm: log 5 25
- Let's represent the base and the argument as a power of five: 5 = 5 1 ; 25 = 52;
- Let's make and solve the equation:
log 5 25 = b ⇒ (5 1) b = 5 2 ⇒ 5 b = 5 2 ⇒ b = 2; - Received an answer: 2.
Task. Calculate the logarithm:
Task. Calculate the logarithm: log 4 64
- Let's represent the base and the argument as a power of two: 4 = 2 2 ; 64 = 26;
- Let's make and solve the equation:
log 4 64 = b ⇒ (2 2) b = 2 6 ⇒ 2 2b = 2 6 ⇒ 2b = 6 ⇒ b = 3; - Received an answer: 3.
Task. Calculate the logarithm: log 16 1
- Let's represent the base and the argument as a power of two: 16 = 2 4 ; 1 = 20;
- Let's make and solve the equation:
log 16 1 = b ⇒ (2 4) b = 2 0 ⇒ 2 4b = 2 0 ⇒ 4b = 0 ⇒ b = 0; - Received a response: 0.
Task. Calculate the logarithm: log 7 14
- Let's represent the base and the argument as a power of seven: 7 = 7 1 ; 14 is not represented as a power of seven, because 7 1< 14 < 7 2 ;
- It follows from the previous paragraph that the logarithm is not considered;
- The answer is no change: log 7 14.
A small note on the last example. How to make sure that a number is not an exact power of another number? Very simple - just decompose it into prime factors. And if such factors cannot be collected in a degree with the same indicators, then the original number is not an exact degree.
Task. Find out if the exact powers of the number are: 8; 48; 81; 35; fourteen.
8 \u003d 2 2 2 \u003d 2 3 is the exact degree, because there is only one multiplier;
48 = 6 8 = 3 2 2 2 2 = 3 2 4 is not an exact power because there are two factors: 3 and 2;
81 \u003d 9 9 \u003d 3 3 3 3 \u003d 3 4 - exact degree;
35 \u003d 7 5 - again not an exact degree;
14 \u003d 7 2 - again not an exact degree;
We also note that we prime numbers are always exact powers of themselves.
Decimal logarithm
Some logarithms are so common that they have a special name and designation.
The decimal logarithm of the x argument is the base 10 logarithm, i.e. the power to which you need to raise the number 10 to get the number x. Designation: lg x .
For example, log 10 = 1; log 100 = 2; lg 1000 = 3 - etc.
From now on, when a phrase like “Find lg 0.01” appears in the textbook, know that this is not a typo. This is the decimal logarithm. However, if you are not used to such a designation, you can always rewrite it:
log x = log 10 x
Everything that is true for ordinary logarithms is also true for decimals.
natural logarithm
There is another logarithm that has its own notation. In a sense, it is even more important than decimal. It's about about the natural logarithm.
The natural logarithm of the argument x is the logarithm to the base e , i.e. the power to which the number e must be raised to obtain the number x. Designation: ln x .
Many will ask: what else is the number e? This is an irrational number, its exact value cannot be found and written down. Here are just the first numbers:
e = 2.718281828459...
We will not delve into what this number is and why it is needed. Just remember that e is the base of the natural logarithm:
ln x = log e x
Thus ln e = 1; log e 2 = 2; ln e 16 = 16 - etc. On the other hand, ln 2 is an irrational number. In general, the natural logarithm of any rational number irrational. Except, of course, unity: ln 1 = 0.
For natural logarithms, all the rules that are true for ordinary logarithms are valid.
In the sixteenth century, navigation developed rapidly. Therefore, the monitoring celestial bodies. To simplify astronomical calculations, in the late 16th - early 17th centuries, logarithmic calculations.
The value of the logarithmic method lies in the reduction of multiplication and division of numbers to addition and subtraction. Less labor intensive activities. Especially if you have to work with multi-digit numbers.
Burgi's method
The first logarithmic tables were compiled by the Swiss mathematician Jost Bürgi in 1590. The essence of his method was as follows.
To multiply, for example, 10,000 by 1,000, it is enough to count the number of zeros in the multiplicand and the multiplier, add them (4 + 3) and write down the product 10,000,000 (7 zeros). Factors are integer powers of 10. When multiplying, the exponents are added. Division is also performed. It is replaced by the subtraction of exponents.
Thus, not all numbers can be divided and multiplied. But there will be more of them if we take a number close to 1 as the base. For example, 1.000001.
This is what the mathematician Jost Bürgi did four hundred years ago. True, his work “Tables of Arithmetic and Geometric, along with a thorough instruction ...” he published only in 1620.
Jost Burgi was born on February 28, 1552 in Liechtenstein. From 1579 to 1604 he served as court astronomer under the Landgrave of Hesse-Kassel Wilhelm IV. Later at Emperor Rudolph II in Prague. A year before his death, in 1631, in Kassel. Bürgi is also known as the inventor of the first pendulum clock.
Napier tables
In 1614 John Napier's tables appeared. This scientist also took as a basis a number close to one. But it was less than one.
Scottish Baron John Napier (1550-1617) studied at home. Loved to travel. Traveled to Germany, France and Spain. At 21, he returned to the family estate near Edinburgh and lived there until his death. Studied theology and mathematics. The latter he studied according to the works of Euclid, Archimedes and Copernicus.
Decimal logarithms
Napier and the Englishman Brigg came up with the idea of compiling a table of decimal logarithms. They began the work of recalculating the previously compiled Napier tables together. After Napier's death, Brigg continued it. He published the work in 1624. Therefore, decimals are also called brigs.
The compilation of logarithmic tables required many years of laborious work from scientists. On the other hand, the labor productivity of thousands of calculators who used the tables compiled by them increased many times over.
