Natural Logarithm

The natural logarithm function can also be defined as the Inverse Function of the Exponential Function , leading to the identities:
:

e^{\ln(x)} = x \qquad \mbox{if }x > 0\,\!

\ln(e^x) = x.\,\!

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.

NOTATIONAL CONVENTIONS

Mathematicians, statisticians, and some engineers generally understand either "log(''x'')" or "ln(''x'')" to mean log_e(''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 "log_e(''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 Language s, including C , C++ , Fortran , and BASIC , "log" or "LOG" refers to the natural logarithm.

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

REASON FOR BEING "NATURAL"

Initially, it seems that in a world using — 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:
:

rac{d}{dx}\log_b(x) =rac{\log_b e}{x}

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.

DEFINITIONS

Formally, ln(''a'') may be defined as the area under the graph ( Integral ) of 1/''x'' from 1 to ''a'', that is,
:

\ln(a)=\int_1^a rac{1}{x}\,dx.

This defines a logarithm because it satisfies the fundamental property of a logarithm:
:

\ln(ab)=\ln(a)+\ln(b) \,\!

This can be demonstrated by letting

t=	frac xa

as follows:

:

\ln (ab) 
= \int_1^{ab} rac{1}{x} \; dx 
= \int_1^a rac{1}{x} \; dx \; + \int_a^{ab} rac{1}{x} \; dx 
=\int_1^{a} rac{1}{x} \; dx \; + \int_1^{b} rac{1}{t} \; dt 
= \ln (a) + \ln (b)

The number '' E '' can then be defined as the unique real number ''a'' such that ln(''a'') = 1.

Alternatively, if the Exponential Function has been defined first using an Infinite Series , the natural logarithm may be defined as its Inverse Function , i.e., ln(''x'') is that function such that

e^{\ln(x)} = x\!

. Since the range of the exponential function on real arguments is all positive real numbers and since the exponential function is strictly increasing, this is well-defined for all positive ''x''.

DERIVATIVE, TAYLOR SERIES

The Derivative of the natural logarithm is given by
:

rac{d}{dx} \ln(x) = rac{1}{x}.\,

This leads to the Taylor Series

This series is similar to a BBP-type Formula .

Also note that

x \over {x-1}

is its own inverse function, so to yield the natural logarithm of a certain number n, simply put in

n \over {n-1}

for x.

THE NATURAL LOGARITHM IN INTEGRATION

:<math>\ {d \over Dx}\left( \ln \left X Ight Ight) {1 \over x}</math>
:<math>\int { Dx \over X} \lnx + C</math>
:<math>\int { Rac{f'(x)}{f(x)}\, Dx} \ln f(x) + C</math>

Letting ''f''(''x'') = cos(''x'') and ''f'''(''x'')= - sin(''x''):

:<math>\ln(1+x) x \,\left( rac{1}{1} - x\,\left(rac{1}{2} - x \,\left(rac{1}{3} - x \,\left(rac{1}{4} - x \,\left(rac{1}{5}- \ldots ight) ight) ight) ight) ight) \quad{ m for}\quad \leftx ight<1\,\!</math>

with ''m'' chosen so that ''p'' bits of precision is attained. In fact, if this method is used, Newton inversion of the natural logarithm may conversely be used to calculate the exponential function efficiently. (The constants ln 2 and π can be pre-computed to the desired precision using any of several known quickly converging series.)

Computational complexity

The Computational Complexity of computing the natural logarithm (using the arithmetic-geometric mean) is O(''M''(''n'') ln ''n''). Here ''n'' is the number of digits of precision at which the natural logarithm is to be evaluated and ''M''(''n'') is the computational complexity of multiplying two ''n''-digit numbers.

COMPLEX LOGARITHMS

See Also: Complex logarithm

The exponential function can be extended to a function which gives a Complex Number as ''e''^''x'' for any arbitrary complex number ''x''; simply use the infinite series with ''x'' complex. This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm. There are two difficulties involved: no ''x'' has ''e''^''x'' = 0; and it turns out that ''e''^2''πi'' = 1 = ''e''⁰. Since the multiplicative property still works for the complex exponential function, ''e''^''z'' = ''e''^{''z''+2''nπi''}, for all complex ''z'' and integers ''n''.

So the logarithm cannot be defined for the whole Complex Plane , and even then it is multi-valued – any complex logarithm can be changed into an "equivalent" logarithm by adding any integer multiple of 2''πi'' at will. The complex logarithm can only be single-valued on the Cut Plane . For example, ln ''i'' = 1/2 ''πi'' or 5/2 ''πi'' or −3/2 ''πi'', etc.; and although ''i''⁴ = 1, 4 log ''i'' can be defined as 2''πi'', or 10''πi'' or −6 ''πi'', and so on.

Image:NaturalLogarithmImpng ''z'' Im(ln(x+iy))
Image:NaturalLogarithmAbspng ''z'' ln(x+iy)
"http://betterexplainedcom/articles/demystifying-the-natural-logarithm-ln/" class="copylinks" target="_blank">Demystifying the Natural Logarithm (ln) BetterExplained

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Natural Logarithm