Natural logarithm

Inverse of exponential

Integral definition

The natural logarithm of a positive, real number a may be defined as the area under the graph of the hyperbola with equation y = 1/x between x = 1 and x = a.

The periodic table was invented by Russian chemist Dmitri Mendeleev in 1869. He was known as a charismatic teacher and lecturer, and held a series of academic and teaching positions throughout the 1860s. Meanwhile, he continued his research. He kept a collection of cards, each of which contained data on a different element. On 17 February 1869, while arranging his cards in order of atomic weight, he suddenly noticed a repeating pattern, whereby elements with similar properties would appear at regular intervals. He had discovered the phenomenon of periodicity, and it was this discovery that led to the formation of the periodic table we know and use today.

The order of the electron orbital energy levels, starting from least to greatest, is as follows: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p.

The number \(e\), also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of natural logarithms. Its value is the limit of \({(1\;+\;1/n)}^n\) as \(n\) approaches infinity.

The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms to the base \(e\). It is assumed that the table was written by William Oughtred.

\(e=\lim\limits_{n\rightarrow\infty}{(1+\frac1n)}^n\)

\(e=1+\frac1{1!}+\frac1{2!}+\frac1{3!}+\frac1{4!}+\frac1{5!}+\cdots\)

Euler's number frequently appears in problems related to growth or decay, where the rate of change is determined by the present value of the number being measured. One example is in biology, where bacterial populations are expected to double at reliable intervals.

\(e^{-x}=1–\frac x{1!\;}+\frac{x^2}{2!\;}–\frac{x^3}{3!\;}+\frac{x^4}{4!\;}–\frac{x^5}{5!\;}+\;\cdots\)

One property that goes to the essence of e and makes it so natural for logarithms and situations of exponential growth and decay is this:

\(\frac{\operatorname d{}}{\operatorname dx}e^x=e^x\)

This says that the rate of change of ex is equal to its value at all points. When x represents time, it signifies a rate of growth (or decay, for negative x) that is equal to the size or quantity that has accumulated thus far.

Euler's Formula

Euler's formula for complex analysis

\(e^{ix}=\cos\left(x\right)+i\sin\left(x\right)\)

${\displaystyle \sin x=\frac{e^{ix}-e^{-ix}}{2i}}$

${\displaystyle \cos x=\frac{e^{ix}+e^{-ix}}{2}}$

${\displaystyle \sinh x=\frac{1}{2}(e^x-e^{-x})}$

${\displaystyle \cosh x=\frac{1}{2}(e^x+e^{-x})}$