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Using simpler operations

Logarithms can be used to make calculations easier. For example, two numbers can be

multiplied just by using a logarithm table and adding.

 because

 becau

se

 because

 becau

se

 becau

se

Where b, x, and y are positive real numbers and . Both c and d are real numbers.

[edit] Trivial identities

 because

 because

  Note that is undefined because there is no number such that . In fact, thereis a vertical asymptote on the graph of at x = 0.

[edit] Canceling exponentials

Logarithms and exponentials (antilogarithms) with the same base cancel each other. This is

true because logarithms and exponentials are inverse operations (just like multiplication and

division or addition and subtraction).

[edit] Changing the base

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This identity is needed to evaluate logarithms on calculators. For instance, most calculatorshave buttons for ln and for log10, but not for log2. To find log2(3), one must calculate log10(3)

/ log10(2) (or ln(3)/ln(2), which yields the same result).

[edit] Proof 

Let c = logba.

Then bc = a.

Take logd  on both sides: logd bc

= logd a 

Simplify and solve for c: clogd b = logd a 

Since c = logba, then

This formula has several consequences:

where is any permutation of the subscripts 1, ..., n. For example

[edit] Summation/subtraction

The following summation/subtraction rule is especially usefulin probability theory when one

is dealing with a sum of log-probabilities:

which gives the special cases:

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 Note that in practice a and c have to be switched on the right hand side of the equations if c 

> a. Also note that the subtraction identity is not defined if a = c since the logarithm of zero

is not defined.

More generally:

where .

[edit] Exponents

A useful identity involving exponents:

[edit] Calculus identities

[edit] Limits 

The last limit is often summarized as "logarithms grow more slowly than any power or root

of  x".

[edit] Derivatives of logarithmic functions

Where x > 0, b > 0, and .

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[edit] Integral definition

[edit] Integrals of logarithmic functions

To remember higher integrals, it's convenient to define:

Then,

[edit] Approximating large numbers

The identities of logarithms can be used to approximate large numbers. Note thatlogb(a) + logb(c) = logb(ac), where a, b, and c are arbitrary constants. Suppose that one wants

to approximate the 44th Mersenne prime, 232,582,657

í 1. To get the base-10 logarithm, wewould multiply 32,582,657 by log10(2), getting 9,808,357.09543 = 9,808,357 + 0.09543. We

can then get 109,808,357

× 100.09543

§ 1.25 × 109,808,357

.

Similarly, factorials can be approximated by summing the logarithms of the terms.

[edit] Complex logarithm identities

The complex logarithm is the complex number analogue of the logarithm function. No singlevalued function on the complex plane can satisfy the normal rules for logarithms. However a

multivalued function can be defined which satisfies most of the identities. It is usual to

consider this as a function defined on aRiemann surface. A single valued version called the

 principal value of the logarithm can be defined which is discontinuous on the negative x axis

and equals the multivalued version on a single branch cut.

[edit] Definitions

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The convention will be used here that a capital first letter is used for the principal value of functions and the lower case version refers to the multivalued function. The single valued

version of definitions and identities is always given first followed by a separate section for the multiple valued versions.

ln(r ) is the standard natural logarithm of the real number r .

Log( z ) is the principal value of the complex logarithm function and has imaginary part in the range (-, ].

Arg( z ) is the principal value of the arg function, its value is restricted to (-, ]. It can be computed using Arg( x+iy)= atan2( y,  x).

The multiple valued version of log( z ) is a set but it is easier to write it without braces and

using it in formulas follows obvious rules.

log( z ) is the set of complex numbers v which satisfy ev

=  z  

arg( z ) is the set of possible values of the arg function applied to z .

When k  is any integer:

log( z ) = ln( |  z  | ) + iarg( z ) 

elog( z ) =  z  

[edit] Constants

Principal value forms:

Multiple value forms, for any k an integer:

log(1) = 0 + 2ik  

log(e) = 1 + 2ik  

[edit] Summation

Principal value forms:

Multiple value forms:

log( z 1) + log( z 2) = log( z 1 z 2) 

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log( z 1) í log( z 2) = log( z 1 /  z 2) 

[edit] Powers

A complex power of a complex number can have many possible values.

Principal value form:

Multiple value forms:

Where k 1, k 2 are any integers: