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Natural logarithm

From Wikipedia, the free encyclopedia

The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828459.

In simple terms, the natural logarithm of a number x is the power to which e would have to be raised to equal x - for example the natural log of e itself is 1 because e1[주: e의 1승] = e, while the natural logarithm of 1 would be 0, since e0[주: e의 0승] = 1 (see the x-intercept of the graph).

The natural logarithm can be defined for all positive real numbers x as the area under the curve y = 1/t from 1 to x, and can also be defined for non-zero complex numbers as explained below.

[그래프 생략]

Graph of the natural logarithm function.

The function goes to negative infinity as x approaches 0, but grows slowly to positive infinity as x increases in value.The natural logarithm function can also be defined as the inverse function of the exponential function, leading to the identities:

In other words, the logarithm function is a bijection from the set of positive real numbers to the set of all real numbers. More precisely it is an isomorphism from the group of positive real numbers under multiplication to the group of real numbers under addition.

Logarithms can be defined to any positive base other than 1, not just e, and are useful for solving equations in which the unknown appears as the exponent of some other quantity.

Contents [hide]

1 Notational conventions

2 Reason for being "natural"

3 Definitions

4 Derivative, Taylor series

5 The natural logarithm in integration

6 Numerical value

6.1 High precision

6.2 Computational complexity

7 Complex logarithms

8 See also

9 External links

10 References

[edit] Notational conventions

Mathematicians, statisticians, and some engineers generally understand either "log(x)" or "ln(x)" to mean loge(x), i.e., the natural logarithm of x, and write "log10(x)" if the base-10 logarithm of x is intended.

Some engineers, biologists, and some others generally write "ln(x)" (or occasionally "loge(x)") when they mean the natural logarithm of x, and take "log(x)" to mean log10(x) or, in the case of some computer scientists, log2(x).

In most commonly-used programming languages, including C, C++, Fortran, and BASIC, "log" or "LOG" refers to the natural logarithm.

In hand-held calculators, the natural logarithm is denoted ln, whereas log is the base-10 logarithm.

[edit] Reason for being "natural" [자연로그라 칭하는 이유-아래에 일부 번역문]

Initially, it seems that in a world using base 10 for nearly all calculations, this base would be more "natural" than base e. The reason we call the ln(x) "natural" is two-fold: first, expressions in which the unknown variable appears as the exponent of e occur much more often than exponents of 10, and second, because the natural logarithm can be defined quite easily using a simple integral or Taylor series - this is not true of other logarithms. Thus, the natural logarithm is more useful in practice. To put it concretely, consider the problem of differentiating a logarithmic function:

If the base b is equal to e then the derivative is simply 1/x, and at x = 1 the slope of the graph is 1.

There are other reasons the natural logarithm is natural; there are a number of simple series involving the natural logarithm, and it often arises in nature. In fact, Nicholas Mercator first described them as log naturalis before calculus was even conceived.

우리가 자연로그라 칭하는 이유는 두 가지이다.

먼저, 미지의 변수들이 10의 지수보다는 e의 지수가 훨씬 자주 나타난다.

그리고, 단순한 적분이나 테일러 시리즈를 이용하여 자연로그가 매우 쉽게 정의되기 때문이다.

그러므로, 자연로그는 실제에서 보다 유용하다.

구체적으로는, 로그함수를 미분하는 문제를 생각해 보기 바란다.

자연로그라 칭하는 또 다른 이유들이 있다. 자연로그를 포함하는 단순한 급수가 많이 있으며, 자연에서 종종 나타난다. 실로, 니콜라스 머케이터는 최초로 그들을 [로그 나투랄리스]라 칭하였는데, 이는 미적분학이 아직 정립되기 전이었다.

[후략]

* 자연로그의 밑수 e를 함께 연구함이 좋겠습니다.