What's the use of natural logarithm?

when x approaches positive infinity or negative infinity, the limit of [1+(1/x)] x is equal to E. In fact, E is discovered through this limit. It is an infinite acyclic decimal. Its value is about equal to 2.718281828...

It is expressed by e

Logarithm with base E is usually used for ㏑

And E is also a transcendental number

e is widely used in science and technology, and generally does not use logarithm with base 1. With E as the base, many formulas can be simplified, and it is the most "natural", so it is called "natural logarithm".

Vortex or spiral is a very common form of natural things, such as: a wisp of smoke curling up into the blue sky, a ripple gently swinging in a blue lake, several snails slowly climbing on the fence and countless stars dancing in the quiet night sky ...

The aesthetic significance of spiral, especially logarithmic spiral, can be expressed in exponential form:

φkρ=αe

. For the convenience of discussion, we define the form of e or by e as "natural law" Therefore, the core of "natural law" is E, and its value is 2.71828 ..., which is an infinite cyclic number.

the beauty of "natural law"

"natural law" is the form of E and E after some transformation and compound. E is the essence of "natural law", and it is a function in mathematics:

(1+1/x) x

the limit when x approaches infinity.

when people study some practical problems, such as the cooling of objects, the reproduction of cells and the decay of radioactive elements, they should all study the x power of

(1+1/x) x

and the limit when x approaches infinity. It is this finiteness obtained from infinite change that develops in two opposite directions (when X tends to positive infinity, the limit of the above formula is equal to E = 2.71828 ... and when X tends to negative infinity, the result of the above formula is equal to E = 2.71828 ...), which fully embodies the most essential things of the formation, development and decline of the universe.

Modern cosmology shows that the universe originated from the "Big Bang" and is still expanding. This description is consistent with the entropy law, one of the two great discoveries in the second half of the 19th century, that is, the second law of thermodynamics. The law of entropy points out that the evolution of matter is always in the direction of destroying information and disintegrating order, and gradually degenerates from complex to simple and from advanced to low. The limit of degradation is the disorderly balance, that is, the state of maximum entropy, a state of inaction and death. What does this process look like? It is not difficult to understand as long as we look at the photos of spiral galaxies in astrophotography. If we must find the kind of motive force that Aristotle said, then we can regard the universe as a clockwork organization that has been tightened in advance, or simply regard the whole universe as a huge clockwork. History is just a process in which this clockwork constantly strives for freedom and releases energy.

The evolution of living things has the opposite characteristics, which tends to be very different from the entropy described by the second law of thermodynamics, so that living matter can avoid the tendency and environmental decline. Any life is a dissipative structure system, and the reason why it can avoid the death state approaching the maximum entropy is because life can continuously absorb negative entropy from the environment through metabolic processes such as eating, drinking and breathing. The essential thing in metabolism is to make the organism successfully eliminate all the entropy that it has to produce when it is alive.

On the one hand, the "law of nature" embodies the collapse process of natural systems in a chaotic direction (such as the decay of elements), on the other hand, it shows the essence that life systems can maintain their own stability and promote their own development (such as cell reproduction) only through an orderly process. It is precisely because of this feature that order and disorder, vitality and death are in the same form that "natural law" has important aesthetic value.

If the barren and boundless desert is the entropy increasing state of "natural law" in disorder and death, then the vast and vibrant grassland is the dynamic and stable structure of "natural law" in order and prosperity. Therefore, the desert makes people feel solemn, boundless and thought-provoking, and reminds people of all kinds of difficulties and frustrations in the course of life; Grassland, on the other hand, makes people excited and happy, and makes people feel the joy and happiness of life.

e = 2.71828 ... is a quantitative expression of "natural law". The image expression of "natural law" is a spiral. There are usually five mathematical expressions of spiral: (1) logarithmic spiral; (2) archimedean spiral; (3) interlocking spiral; (4) Hyperbolic spiral; (5) spiral. Logarithmic spiral is the most common in nature, and other spirals are also related to logarithmic spiral, but at present we have not found the general formula of spiral. The logarithmic spiral was introduced by Descartes in 1638. Later, the Swiss mathematician Jacob Bernoulli studied it in detail, and found that the involute and extension lines of the logarithmic spiral are still logarithmic spiral, and the tangents of the poles at each point of the logarithmic spiral are still logarithmic spiral, and so on. Bernoulli was amazed at these interesting properties and left a will to draw the logarithmic spiral on his tombstone.

Willianm Hogarth, a famous British painter and art theorist, deeply felt that spiral or spiral shapes gradually shrink to their centers, and they are all beautiful shapes. In fact, it is easy for us to find the spiral in the works of ancient and modern art masters. Why can our feelings and our "spiritual" eyes often be satisfied instinctively and intuitively from such a spiral form? Doesn't this mean that there is a more primitive isomorphic correspondence between our spirit, our "inner" world and the external world than history?

As we know, protein, as the basic substance of life phenomenon, participates in the whole work of life process in living objects, and its function is so complicated, efficient and mysterious that it is closely related to its structure. Chemists found that protein's titanium chain is mainly spiral, and the structure of nucleic acid, which determines heredity, is also spiral.

The ancient Greeks had a musical instrument called Feng Ming Qin. When its strings vibrated in the wind, it could produce beautiful and pleasant tones. This tone is the so-called "vortex wake effect". It is thought-provoking that the inner ear structure of the hearing organ evolved by human beings over a long period of time also has a vortex shape. Is this for the convenience of enjoying the ancient Greek wind chime? There are also our fingerprints, hairpins, etc. The physiological structure of this aesthetic subject corresponds to the isomorphism of the external world, which is the natural basis for the harmony between "inner" and "outer".

Some people say that the beauty of mathematics is the brilliance of "one", which has the invariance under the action of as many transformation groups as possible, that is, it has the expression of the common laws of nature and is the unity of "many" and "one", so the "natural law" also flashes the brilliance of "one". Who can tell how much convenience and success E = 2.71828 ... has brought to mathematicians? People praise the rigidity, clarity and frankness of straight lines, appreciate the beauty, change and implication of curves, but they don't know that any straight line and curve can be composed of enough parts from the spiral. Some people say that beauty is the unity of subject and object, and the unity of inner spiritual world and outer material world, so the "natural law" also has this unity. Human understanding develops according to the law of negation of negation, and the history of society and nature also follows this dialectical development law. What gives this form a vivid expression? Spiral!

Some people say that beauty lies in the rhythm of things, and the "natural law" also has this rhythm; Some people say that beauty is a dynamic balance and a changing eternity, so "natural law" is also a dynamic balance and a changing eternity; Some people say that beauty lies in the dynamic structure of things, so the "law of nature" also has this structure-such as the hairspring in the watch, the spring in the machine and so on.

"natural law" is the unity of formal cause and dynamic cause, the image manifestation of things, and the same expression of concrete and abstract. Limited life is rooted in infinite nature, and the pulse of life consciously adjusts the movement and rhythm according to the melody of the universe ... Organic and inorganic, internal and external, social and natural, and everything is integrated. Is this all the aesthetic mystery revealed by the "natural law"? No! The "natural law" always has inexhaustible aesthetic connotation, because it symbolizes the vast and profound nature. Because of this, it attracts people who are worthy to make unremitting exploration, thus showing the essential power of human evolution. (Originally published in Science Spring magazine, No.4, 1984, The original title is "Natural Laws-Treasures of Aesthetists and Artists")

Attachment:

This is two thousand digits after the decimal point:

E =: 2.71828 18284 5945 23536 2874 71352 66249 77572 4793 69995 95749 66967. 13 82178 52516 64274 27466 39193 23 59921 81741 35966 2943 5729 3342 9526 59563 7381 32328 62794 3497 63233 82988 7531 95251 191 15738 3418 7 937 2154 89149 93488 41675 9244 76146 668 82264 816 84774 11853 74234 54424 3717 5397 77449 9269 5517 27618 3866 26133 13845 83 7524 49338 2656 2976 67371 132 7932 8791 27443 7474 7236 96977 2931 1416 92836 8192 55151 8657 46377 21112 52389 78442 5569 53696 7778 54499 69967 94686 44549 5987 93163 68892 398 79312 77361 78215 42499 92295 76351 4822 82698 95193 6683 31825 28869 39849 64651 582 93923 98294 88793 3 236 2594 43117 3123 8197 68416 1439 7198 37679 3268 32823 76464 8429 53118 2328 7825 98194 55815 3175 67173 61332 6981 1259 96181 88159 3 416 9351 59888 85193 4587 27386 67385 89422 87922 84998 9286 8582 57492 7961 48419 84443 63463 24496 84875 6233 62482 7419 78623 29 2169 92 35 3436 99418 49146 3149 34317 38143 6454 62531 5296 18369 8887 716 76839 64243 7814 59271 45635 4961 331 7285 1383 7551 1157 4774 1718 9 8616 87396 96552 12671 54688 9573 5354 2123 4784 98193 34321 6817 121 5627 8823 5193 33224 7451 58539 473 41995 7777 9353 6641 69973 29725 8868 76966 4355 5771 62268 44716 2567 98826 51787 13419 51246 6521 359 21236 67719 43252 78675 39855 89448 9697 9649 75459 18569 5638 23637 1621 1247 74272 28364 89613 42251 6445 78182 44235 29486 36372 14174 2388 93441 24796 35743 7263 75529 44483 37998 1612 54922 7859 25778 2 562 92622 64832 62779 33386 56648 16277 25164 191 594 91644 99828 9315 5664 7258 27786 31864 15519 56532 44258 69829 46959 381 91529 87211 72 556 34754 63964 4791 1459 495 86298 49679 12874 687 5489 58586 71747 98546 67757 5732 56812 88459 2541 3345 3922 113 7863 9455 6688 166 74 169 8425 584 33637 95376 4523 424 32256 61352 78369 51177 88386 38744 39662 53224 9856 54995 88623 42818 9977 73327 61717 83928 3494 651 4 34558 8977 19425 86398 77275 4719 62953 74152 11151 36835 6275 2623 26484 7287 3927