Translation，lnx =(x- 1)-(x- 1)2/2+(x- 1)3/3-(x- 1)4/4+...+(-65438)

So lnx

Let F(X) = f (x)-inx, that is, it is proved that the function F(X) is within x (1, e) and the equation F(x)=0 has a solution.

The derivative of f(x) is f' (x) = f' (x)- 1/x, and because the function F(x) is derivable on [1, e], 0

F '(x)≦( 1/x), so f' (x) >1; 0[ 1, e] is a constant, so F(x) decreases monotonically, F( 1)=f( 1)>0, f (e) = f (x)-1<; 0。

So there is a unique solution: F(x)=0, that is, there is only one x in (1, e), so f (x) = lnx.

Extended data:

Mathematics pays attention to regularity and aesthetics, but the basic constants such as pi and natural logarithm e are so chaotic, just like two "mathematical ghosts". People can't find the law of the number change of π and E. The possible reasons are: for example, people used decimal system, and the ancients used hand index because it was ten fingers.

So the decimal system is determined, and binary system is the simplest system in the universe, which also conforms to the theory of Yin and Yang. 1 is positive and 0 is negative. For another example, when people compare π and E with those regular numbers, they think that E and π are disordered, so the problem of "reference" is involved. So, if you convert π and E into the simplest binary.

Comparing the two chaotic numbers π and E with each other, we will find some numerical laws. The first 17 bit of the fractional part of e and the 5th-21bit of the fractional part of π are exactly in reverse order. It may not be a coincidence that the order is so long.

Baidu encyclopedia-natural logarithm