What is the logarithm?

The concept of logarithm

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English noun: logarithm

If a b = n, then log (a) (n) = B. Where a is called "base", n is called "true number" and b is called "logarithm of n with base A".

The log(a)(n) function is called logarithmic function. The domain of X in logarithmic function is x>0, and zero and negative numbers have no logarithm. The domain of A is a>0 and a ≠ 1.

Logarithmic history

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Logarithm is an important content in elementary mathematics in middle school, so who initiated the advanced operation of logarithm at the beginning? In the history of mathematics, it is generally believed that the inventor of logarithm is the Scottish mathematician-Baron Napier (Napier, 1550- 16 17). In Napier's time, Copernicus's "sun-centered theory" was just popular, which led to astronomy becoming a hot subject at that time. However, due to the limitations of constant mathematics at that time, astronomers had to spend a lot of energy to calculate those complicated "astronomical figures", thus wasting precious time for several years or even a lifetime. Napier was also an astronomy enthusiast at that time. In order to simplify the calculation, he devoted himself to studying the calculation technology of large numbers for many years, and finally invented logarithm independently. Of course, the logarithm invented by Napier is not exactly the same as the logarithm theory in modern mathematics. In Napier's time, the concept of "exponent" had not been formed, so Napier did not derive logarithm by exponent as in the algebra textbooks now, but obtained the concept of logarithm by studying linear motion. So, what about the logarithmic operation invented by Napier at that time? At that time, it was still a very complicated operation to calculate the product between multiple digits, so Napier first invented a method to calculate the product between special multiple digits. Let's look at the following example:

n 0、 1、2、3、4、5、6、7、8、9、 10、 16、 12、 13、 14……

2^n 1、2、4、8、 16、32、64、 128、256、5 12、2048、4096、8 192、 16384、……

The relationship between these two lines is extremely clear: the first line represents the exponent of 2, and the second line represents the corresponding power of 2. If we want to calculate the product of two numbers in the second row, we can add up the corresponding numbers in the first row. For example, to calculate the value of 64×256, you can first query the number corresponding to the first line: 64 corresponds to 6, and 256 corresponds to 8; Then add up the numbers corresponding to the first line: 6+8 =14; The first line 14 corresponds to the second line 16384, so there is: 64× 256 = 16384. Napier's calculation method is actually the idea of "logarithmic operation" in modern mathematics. Looking back, when we were studying "Simplifying Calculation with Logarithms" in middle school, didn't we adopt this idea: to calculate the product of two complex numbers, first look up the common logarithm table to find out the common logarithm of these two complex numbers, then add these two common logarithms, and then find out the opposite value of the added sum through the opposite table of common logarithms, which is the product of the original two complex numbers. Isn't this idea of "changing multiplication and division into addition and subtraction" to simplify calculation an obvious feature of logarithmic operation? After many years' exploration, Baron Napier published his masterpiece "Interpretation of Logarithmic Wonder Law" in 16 14, announced his invention to the whole world, and expounded its characteristics. Therefore, Napier is a well-deserved "logarithmic creator" and is worthy of this honor in the history of mathematics. In the book Dialectics of Nature, Engels, the great tutor, once called Cartesian coordinates, Napier logarithm, Newton calculus and Leibniz calculus as1the three major mathematical inventions in the 7th century. Laplace (1749- 1827), a famous French mathematician and astronomer, once said that logarithm can shorten the calculation time, "in fact, it is equivalent to prolonging the life span of astronomers many times".

Properties and Derivation of Logarithm

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Definition:

If a n = b(a >；; 0 and a ≠ 1)

Then n=log(a)(b)

Basic nature:

1、a^(log(a)(b))=b

2、log(a)(MN)= log(a)(M)+log(a)(N)；

3、log(a)(M÷N)= log(a)(M)-log(a)(N)；

4、log(a)(M^n)=nlog(a)(M)

infer

1, because n=log(a)(b) and a n = b, that is, a (log (a) (b)) = b.

2、MN=M×N

By the basic properties of 1 (replacing m and n)

a^[log(a)(mn)]= a^[log(a)(m)]×a^[log(a)(n)]

According to the nature of the index

a^[log(a)(mn)]= a^{[log(a)(m)]+[log(a)(n)]]

And because exponential function is monotone function, so

log(a)(MN) = log(a)(M) + log(a)(N)

3, similar to (2)

MN=M÷N

By the basic properties of 1 (replacing m and n)

a^[log(a)(m÷n)]= a^[log(a)(m)]÷a^[log(a)(n)]

According to the nature of the index

a^[log(a)(m÷n)]= a^{[log(a)(m)]-[log(a)(n)]]

And because exponential function is monotone function, so

log(a)(M÷N) = log(a)(M) - log(a)(N)

4, similar to (2)

M^n=M^n

From the basic attribute 1 (replace m)

a^[log(a)(m^n)]= {a^[log(a)(m)]}^n

According to the nature of the index

a^[log(a)(m^n)]= a^{[log(a)(m)]*n}

And because exponential function is monotone function, so

log(a)(M^n)=nlog(a)(M)

Basic Attribute 4 Summary

log(a^n)(b^m)=m/n*[log(a)(b)]

Derived as follows:

According to the formula (see below) [lnx is the base of log(e)(x)e called natural logarithm] log (an n log (a n) (b m) = ln (a n) ÷ ln (b n).

It can be obtained from basic attribute 4.

log(a^n)(b^m) = [n×ln(a)]

Then according to the bottom changing formula

log(a^n)(b^m)=m÷n×[log(a)(b)]

Functional image

1. Logarithmic function images all pass through the (1, 0) point.

2. For the function y=log(a)(n),

①, when 0

2 When a> is at 1, the display function on the image is (0, +∞) increasing. With the increase of a, the image gradually rotates counterclockwise around the point (1.0), but it does not exceed X= 1.

3. Similar to the mirror image relationship between other functions and inverse functions, the mirror image of logarithmic function and exponential function is symmetrical about the straight line y = x. 。

Other attributes

Attribute 1: bottoming formula

log(a)(N)=log(b)(N)÷log(b)(a)

Derived as follows:

N = a^[log(a)(N)]

a = b^[log(b)(a)]

By combining the two formulas, it can be concluded that.

n = {b^[log(b)(a)]}^[log(a)(n)]= b^{[log(a)(n)]*[log(b)(a)]}

And because n = b [log (b) (n)]

So b [log (b) (n)] = b {[log (a) (n)] * [log (b) (a)]}

So log (b) (n) = [log (a) (n)] * [log (b) (a)] {If you don't understand this step or have questions, please see above}

So log(a)(N)=log(b)(N)/log(b)(a)