# Logarithm and Natural Logarithm

The *base* b *logarithm* of a number is the *exponent* that we need to raise the *base* in order to get the number.

Logarithm definition

When b is raised to the power of y is equal x:

*b ^{ y}* =

*x*

Then the base b logarithm of x is equal to y:

log* _{b}*(

*x*)

*= y*

For example when:

2^{4} = 16

Then

log_{2}(16) = 4

### Logarithm as inverse function of exponential function

The logarithmic function,

*y *= log* _{b}*(

*x*)

is the inverse function of the exponential function,

*x *=* b ^{y}*

So if we calculate the exponential function of the logarithm of x (x>0),

*f *(*f *^{-1}(*x*)) = *b*^{log}*b*^{(x)} = *x*

Or if we calculate the logarithm of the exponential function of x,

*f *^{-1}(*f *(*x*)) = log_{b}(*b ^{x}*) =

*x*

### Natural logarithm (ln)

Natural logarithm is a logarithm to the base e:

ln(*x*) = log* _{e}*(

*x*)

When e constant is the number:

### Inverse logarithm calculation

The inverse logarithm (or anti logarithm) is calculated by raising the base b to the logarithm y:

*x* = log^{-1}(*y*) = *b ^{ y}*

### Logarithmic function

The logarithmic function has the basic form of:

*f *(*x*) = log* _{b}*(

*x*)

Logarithm rules

**Rule name**

**Rule**

*(*

_{b}*x ∙ y*) = log

*(*

_{b}*x*)

*+*log

*(*

_{b}*y*)

*(*

_{b}*x / y*) = log

*(*

_{b}*x*)

*–*log

*(*

_{b}*y*)

*(*

_{b}*x*) =

^{y}*y ∙*log

*(*

_{b}*x*)

*(*

_{b}*c*) = 1 / log

*(*

_{c}*b*)

*(*

_{b}*x*) = log

*(*

_{c}*x*) / log

*(*

_{c}*b*)

*f*(

*x*) = log

_{b}(

*x*)⇒

*f ‘*(

*x*) = 1 / (

*x*ln(

*b*) )

*(*

_{b}*x*)

*dx*=

*x ∙*( log

*(*

_{b}*x*)- 1 / ln(

*b*)) +

*C*

_{b}(

*x*)is undefined when

*x*≤ 0

_{b}(0) is undefined

_{b}(1) = 0

_{b}(

*b*) = 1

_{b}(

*∞*) =

*∞,*when

*x*→∞

Logarithm product rule

The logarithm of the multiplication of x and y is the sum of logarithm of x and logarithm of y.

log* _{b}*(

*x ∙ y*) = log

*(*

_{b}*x*)

*+*log

*(*

_{b}*y*)

For example:

log_{10}(3* ∙ *7) = log_{10}(3)* + *log_{10}(7)

Logarithm quotient rule

The logarithm of the division of x and y is the difference of logarithm of x and logarithm of y.

log* _{b}*(

*x / y*) = log

*(*

_{b}*x*)

*–*log

*(*

_{b}*y*)

For example:

log_{10}(3* / *7) = log_{10}(3)* – *log_{10}(7)

Logarithm power rule

The logarithm of x raised to the power of y is y times the logarithm of x.

log* _{b}*(

*x*) =

^{y}*y ∙*log

*(*

_{b}*x*)

For example:

log_{10}(2^{8}) = 8*∙ *log_{10}(2)

Logarithm base switch rule

The base b logarithm of c is 1 divided by the base c logarithm of b.

log* _{b}*(

*c*) = 1 / log

*(*

_{c}*b*)

For example:

log_{2}(8) = 1 / log_{8}(2)

Logarithm base change rule

The base b logarithm of x is base c logarithm of x divided by the base c logarithm of b.

log* _{b}*(

*x*) = log

*(*

_{c}*x*) / log

*(*

_{c}*b*)

For example, in order to calculate log_{2}(8) in calculator, we need to change the base to 10:

log_{2}(8) = log_{10}(8) / log_{10}(2)

Logarithm of negative number

The base b real logarithm of x when x<=0 is undefined when x is negative or equal to zero:

log_{b}(*x*) is undefined when *x* ≤ 0

Logarithm of 0

The base b logarithm of zero is undefined:

log_{b}(0) is undefined

The limit of the base b logarithm of x, when x approaches zero, is minus infinity:

Logarithm of 1

The base b logarithm of one is zero:

log_{b}(1) = 0

For example, teh base two logarithm of one is zero:

log_{2}(1) = 0

Logarithm of infinity

The limit of the base b logarithm of x, when x approaches infinity, is equal to infinity:

lim log_{b}(*x*) = ∞, when *x*→∞

Logarithm of the base

The base b logarithm of b is one:

log_{b}(*b*) = 1

For example, the base two logarithm of two is one:

log_{2}(2) = 1

Logarithm derivative

When

*f *(*x*) = log* _{b}*(

*x*)

Then the derivative of f(x):

*f ‘ *(*x*) = 1 / (* x* ln(*b*) )

Logarithm integral

The integral of logarithm of x:

∫log* _{b}*(

*x*)

*dx*=

*x ∙*( log

*(*

_{b}*x*)- 1 / ln(

*b*)) +

*C*

For example:

∫log_{2}(*x*) *dx* = *x ∙ *( log_{2}(*x*)- 1 / ln(2)) + *C*

Natural logarithm is the logarithm to the base e of a number.

Definition of natural logarithm

When

*e ^{ y}* =

*x*

Then base e logarithm of x is

ln(*x*) = log_{e}(*x*)* = y*

The e constant or Euler’s number is:

*e* ≈ 2.71828183

### Ln as inverse function of exponential function

The natural logarithm function ln(x) is the inverse function of the exponential function e^{x}.

For x>0,

*f *(*f *^{-1}(*x*)) = *e*^{ln(x)} = *x*

Or

*f *^{-1}(*f *(*x*)) = ln(*e ^{x}*) =

*x*

Natural logarithm rules and properties

**Rule name**

**Rule**

**Example**

*x ∙ y*) = ln(

*x*)

*+*ln(

*y*)

*∙*7) = ln(3)

*+*ln(7)

*x / y*) = ln(

*x*)

*–*ln(

*y*)

*/*7) = ln(3)

*–*ln(7)

*x*) =

^{y}*y ∙*ln(

*x*)

^{8}) = 8

*∙*ln(2)

*f*(

*x*) = ln(

*x*)⇒

*f ‘*(

*x*) = 1 /

*x*

*x*)

*dx*=

*x ∙*(ln(

*x*) – 1) +

*C*

*x*) is undefined when

*x*≤ 0

*x*) = ∞ ,when

*x*→∞

#### Logarithm product rule

The logarithm of the multiplication of x and y is the sum of logarithm of x and logarithm of y.

log* _{b}*(

*x ∙ y*) = log

*(*

_{b}*x*)

*+*log

*(*

_{b}*y*)

For example:

log_{10}(3* ∙ *7) = log_{10}(3)* + *log_{10}(7)

Logarithm quotient rule

The logarithm of the division of x and y is the difference of logarithm of x and logarithm of y.

log* _{b}*(

*x / y*) = log

*(*

_{b}*x*)

*–*log

*(*

_{b}*y*)

For example:

log_{10}(3* / *7) = log_{10}(3)* – *log_{10}(7)

Logarithm power rule

The logarithm of x raised to the power of y is y times the logarithm of x.

log* _{b}*(

*x*) =

^{y}*y ∙*log

*(*

_{b}*x*)

For example:

log_{10}(2^{8}) = 8*∙ *log_{10}(2)

Derivative of natural logarithm

The derivative of the natural logarithm function is the reciprocal function.

When

*f *(*x*) = ln(*x*)

The derivative of f(x) is:

*f ‘ *(*x*) = 1 / *x*

Integral of natural logarithm

The integral of the natural logarithm function is given by:

When

*f *(*x*) = ln(*x*)

The integral of f(x) is:

∫* f *(*x*)*dx* = ∫ln(*x*)*dx* = *x ∙ *(ln(*x*) – 1) + *C*

Ln of 0

The natural logarithm of zero is undefined:

ln(0) is undefined

The limit near 0 of the natural logarithm of x, when x approaches zero, is minus infinity:

Ln of 1

The natural logarithm of one is zero:

ln(1) = 0

Ln of infinity

The limit of natural logarithm of infinity, when x approaches infinity is equal to infinity:

lim ln(*x*) = ∞, when *x*→∞