logarithm

[duì shù]

Mathematical terminology

Collection

zero Useful+1

zero

This entry is made by Compilation and application of scientific encyclopedia entries of "Science Popularization China" to examine.

In mathematics, logarithms are exponentiated Inverse operation , just as division is the inverse of multiplication, and vice versa.^[6] This means that the logarithm of one number is the exponent that must produce another fixed number (base). In a simple case, the logarithmic count factor in the multiplier. More generally, Power It is allowed to increase any positive real number to any real number, which always produces a positive result. Therefore, you can calculate the logarithm of any two positive real numbers, b and x, where b is not equal to 1.

If the x power of a is equal to N (a>0, and a ≠ 1), then the number x is called the logarithm with a as the base N (logarithm), which is recorded as x=log a N。 Where a is called logarithmic base number , N is called True number 。^[1]

Chinese name: logarithm
Foreign name: logarithm
Common logarithm: Logarithm with base 10, recorded as lgN

Natural logarithm: with Irrational number E is the base logarithm, recorded as lnN
Logarithmic function: Function y=log a x
inventor: Scottish mathematician John Napier
Type: Mathematical terminology

catalog

▪ Basic properties of function

definition

Announce

edit

, i.e a Of x Power equals N （ a >0, and a ≠ 1), then the number x Is called by a Bottom N Logarithm, recorded as

。 Among them, a Called logarithmic base number ， N be called True number ， x It's called "by a Bottom N Of logarithm ”。

one
In particular, the logarithm with base 10 is called Common logarithm (common logarithm) and recorded as lg N 。
two
Called Irrational number e（ e =2.71828...) is called Natural logarithm (natural logarithm) and recorded as ln N 。
three
Zero has no logarithm.^[2]
four
stay real number In the range, negative numbers have no logarithm.^[3] stay imaginary number Within the scope, negative It's logarithmic.

In fact, when

，

, there is e (2 k +1)πi +1=0, so ln (- 1) has multiple periodic values, ln (- 1)=(2 k +1)πi。 In this way, the natural logarithm of any negative number has multiple periodic values. For example: ln (- 5)=(2 k +1)πi+ln 5。^[4]

application

Announce

edit

Logarithms have many applications both inside and outside mathematics. Some of these events are related to the concept of scale invariance. For example, each chamber of the Nautilus shell is an approximate copy of the next, scaled by a constant factor. This leads to a logarithmic spiral. Benford's law on the distribution of leading numbers can also be explained by scale invariance. Logarithms are also related to self similarity. For example, the logarithmic algorithm appears in algorithm analysis, which is solved by decomposing the algorithm into two similar smaller problems and repairing their solutions. The size of self similar geometric shape, that is, the shape of its part similar to the overall image, is also based on logarithm. Logarithmic scales are useful for quantifying relative changes in values opposite to their absolute differences. In addition, since the logarithmic function log (x) grows very slowly for large x, the logarithmic scale is used to compress large-scale scientific data. Logarithms also appear in many scientific formulas, such as Tsiolkovsky rocket equation, Fenske equation or Nernst equation 。^[5]

history

Announce

edit

Napier J.

16. At the turn of the 17th century, with the development of astronomy, navigation, engineering, trade and military, improving the digital calculation method has become an urgent task. John Napier (J. Napier, 1550~1617) It was in the process of studying astronomy that logarithm was invented in order to simplify the calculation. The invention of logarithm is a major event in the history of mathematics, and the astronomical community welcomed this invention with an almost ecstatic mood. Engels The invention of logarithm and the founding of analytic geometry Calculus The establishment of, Galileo He also said: "Give me space, time and logarithm, and I can create a universe."

Before the invention of logarithm trigonometric function The method of transforming the product of into the sum or difference of trigonometric functions is very familiar, and the German mathematician M. Stifel (about 1487~1567) elaborated a corresponding relationship as follows in Comprehensive Arithmetic (1544):

This relationship can be summarized as

At the same time, the operational nature of this relationship (that is, the multiplication, division, power, and derivation of the above line of numbers correspond to the addition, subtraction, multiplication, and division of the following line of numbers) has also been widely known. After years of research on the operation system, Napier published the Instruction to the Wonderful Law of Logarithm in 1614, in which he explained the logarithmic method in geometric terms with the help of kinematics.

It was Napier's friend who transformed the logarithm to make it widely spread Briggs (H. Briggs, 1561~1631), he found it inconvenient to use the logarithm in the Magic Logarithmic Law Instruction, so he agreed with Napier to make the logarithm of 1 0 and the logarithm of 10 1, thus obtaining the common logarithm based on 10. Because of the Number system It is decimal, so it has advantages in numerical calculation. In 1624, Briggs published Logarithmic Arithmetic, publishing 14 digits with a base of 10, including 1~20000 and 90000~100000 Common logarithm table 。

According to the principle of logarithmic operation, people also invented the logarithmic slide rule. For more than 300 years, Logarithmic slide rule It has always been a necessary computing tool for scientific workers, especially engineering technicians, until the 1970s when it gave way to Electronic calculator 。 Although as a calculation tool, logarithmic slide rule Logarithmic table Both are no longer important, but the logarithmic method of thinking still has vitality.

In the process of the invention of logarithm, Napier did not use the reciprocal relationship between exponent and logarithm when discussing the concept of logarithm. The main reason for this is that there was no clear concept of exponent at that time, and even the symbol of exponent was not introduced by French mathematicians until 1637, more than 20 years later Descartes (R. Descartes, 1596~1650). It was not until the 18th century that Swiss mathematicians Euler The inverse relationship between exponent and logarithm is found. In a book published in 1770, Euler first used to define

He pointed out that:“ Logarithm derived from exponent ”。 The invention of logarithm precedes that of index and becomes a treasure in the history of mathematics.

From the invention process of logarithm, we can see that the needs of social production and science and technology are the main driving force for the development of mathematics. The process of establishing the relationship between logarithm and exponent shows that the use of a better symbol system is crucial to the development of mathematics. Actually, okay Mathematical symbol It can greatly save people's thinking burden. Mathematicians have made long-term and arduous efforts to develop and improve the system of mathematical symbols^[2] 。

Symbol

Announce

edit

with a Bottom N Logarithmic notation of

。 The logarithmic sign log comes from the Latin logarithm, which was originally written by Italian mathematicians Kavalieri (Cavalieri). At the beginning of the 20th century, the modern representation of logarithms was formed. For the convenience of use, people gradually record the common logarithm based on 10 and the natural logarithm based on irrational number e as lg N And ln N 。