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:::к№ҖмӨҖнҡЁ ліҖнҳёмӮ¬ нҷҲнҺҳмқҙм§Җм—җ мҳӨмӢ кІғмқ„ нҷҳмҳҒн•©лӢҲлӢӨ:::            к№ҖмӨҖнҡЁ | 2007В·07В·07 22:32 | HIT : 14,207 |    Natural logarithm

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[мЈј: eмқҳ 1мҠ№] = e, while the natural logarithm of 1 would be 0, since e0[мЈј: eмқҳ 0мҠ№] = 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.

[к·ёлһҳн”„ мғқлһө]
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
10 References

 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.

 Reason for being "natural" [мһҗм—°лЎңк·ёлқј м№ӯн•ҳлҠ” мқҙмң -м•„лһҳм—җ мқјл¶Җ лІҲм—ӯл¬ё]

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.

мҡ°лҰ¬к°Җ мһҗм—°лЎңк·ёлқј м№ӯн•ҳлҠ” мқҙмң лҠ” л‘җ к°Җм§ҖмқҙлӢӨ.
лЁјм Җ, лҜём§Җмқҳ ліҖмҲҳл“Өмқҙ 10мқҳ м§ҖмҲҳліҙлӢӨлҠ” eмқҳ м§ҖмҲҳк°Җ нӣЁм”¬ мһҗмЈј лӮҳнғҖлӮңлӢӨ.
к·ёлҰ¬кі , лӢЁмҲңн•ң м Ғл¶„мқҙлӮҳ н…Ңмқјлҹ¬ мӢңлҰ¬мҰҲлҘј мқҙмҡ©н•ҳм—¬ мһҗм—°лЎңк·ёк°Җ л§Өмҡ° мүҪкІҢ м •мқҳлҗҳкё° л•Ңл¬ёмқҙлӢӨ.
к·ёлҹ¬лҜҖлЎң, мһҗм—°лЎңк·ёлҠ” мӢӨм ңм—җм„ң ліҙлӢӨ мң мҡ©н•ҳлӢӨ.
кө¬мІҙм ҒмңјлЎңлҠ”, лЎңк·ён•ЁмҲҳлҘј лҜёл¶„н•ҳлҠ” л¬ём ңлҘј мғқк°Ғн•ҙ ліҙкё° л°”лһҖлӢӨ.
мһҗм—°лЎңк·ёлқј м№ӯн•ҳлҠ” лҳҗ лӢӨлҘё мқҙмң л“Өмқҙ мһҲлӢӨ. мһҗм—°лЎңк·ёлҘј нҸ¬н•Ён•ҳлҠ” лӢЁмҲңн•ң кёүмҲҳк°Җ л§Һмқҙ мһҲмңјл©°, мһҗм—°м—җм„ң мў…мў… лӮҳнғҖлӮңлӢӨ. мӢӨлЎң, лӢҲмҪңлқјмҠӨ лЁёмјҖмқҙн„°лҠ” мөңмҙҲлЎң к·ёл“Өмқ„ [лЎңк·ё лӮҳнҲ¬лһ„лҰ¬мҠӨ]лқј м№ӯн•ҳмҳҖлҠ”лҚ°, мқҙлҠ” лҜём Ғл¶„н•ҷмқҙ м•„м§Ғ м •лҰҪлҗҳкё° м „мқҙм—ҲлӢӨ.

[нӣ„лһө]

* мһҗм—°лЎңк·ёмқҳ л°‘мҲҳ eлҘј н•Ёк»ҳ м—°кө¬н•Ёмқҙ мўӢкІ мҠөлӢҲлӢӨ.       мӢӯм§„лІ• к№ҖмӨҖнҡЁ 07В·07В·17 4837
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