# Logarithm Rules

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
- Logarithm rules
- Logarithm problems
- Complex logarithm
- Graph of log(x)
- Logarithm table
- Logarithm calculator

## 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:

or

See: Natural logarithm

## 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 |
---|---|

## Logarithm product rule |
log(_{b}x ∙ y) = log(_{b}x) + log(_{b}y) |

## Logarithm quotient rule |
log(_{b}x / y) = log(_{b}x) - log(_{b}y) |

## Logarithm power rule |
log(_{b}x ) = ^{y}y ∙ log(_{b}x) |

## Logarithm base switch rule |
log(_{b}c) = 1 / log(_{c}b) |

## Logarithm base change rule |
log(_{b}x) = log(_{c}x) / log(_{c}b) |

## Derivative of logarithm |
f (x) = log_{b}(x) ⇒ f ' (x) = 1 / ( x ln(b) ) |

## Integral of logarithm |
∫ log(_{b}x) dx = x ∙ ( log(_{b}x) - 1 / ln(b) ) + C |

## Logarithm of negative number |
log_{b}(x) is undefined when x≤ 0 |

## Logarithm of 0 |
log_{b}(0) is undefined |

## Logarithm of 1 |
log_{b}(1) = 0 |

## Logarithm of the base |
log_{b}(b) = 1 |

## Logarithm of infinity |
lim log_{b}(x) = ∞,when x→∞ |

See: Logarithm rules

#### 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)

See: log base change rule

#### 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:

See: log of zero

#### 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

See: log of one

#### 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*→∞

See: log of infinity

#### 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*) )

See: log derivative

#### 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*

## Logarithm approximation

log_{2}(*x*) ≈ *n* + (*x*/2^{n} - 1) ,

## Complex logarithm

For complex number z:

*z = re ^{iθ} = x + iy*

The complex logarithm will be (n = ...-2,-1,0,1,2,...):

Log *z = *ln(*r*) + *i*(*θ+2nπ*)* = *ln(√(*x*^{2}+*y*^{2})) + *i*·arctan(*y/x*))

## Logarithm problems and answers

#### Problem #1

Find x for

log_{2}(*x*) + log_{2}(*x*-3) = 2

##### Solution:

Using the product rule:

log_{2}(*x∙*(*x*-3)) = 2

Changing the logarithm form according to the logarithm definition:

*x∙*(*x*-3) = 2^{2}

Or

*x*^{2}-3*x*-4 = 0

Solving the quadratic equation:

*x*_{1,2} = [3±√(9+16) ] / 2 = [3±5] / 2 = 4,-1

Since the logarithm is not defined for negative numbers, the answer is:

*x* = 4

#### Problem #2

Find x for

log_{3}(*x*+2) - log_{3}(*x*) = 2

##### Solution:

Using the quotient rule:

log_{3}((*x*+2) /* x*) = 2

Changing the logarithm form according to the logarithm definition:

(*x*+2)/*x* = 3^{2}

Or

*x*+2 = 9*x*

Or

8*x* = 2

Or

*x* = 0.25

## Graph of log(x)

log(x) is not defined for real non positive values of x:

## Logarithms table

x |
log_{10}x |
log_{2}x |
log_{e}x |
---|---|---|---|

0 | undefined | undefined | undefined |

0^{+} |
- ∞ | - ∞ | - ∞ |

0.0001 | -4 | -13.287712 | -9.210340 |

0.001 | -3 | -9.965784 | -6.907755 |

0.01 | -2 | -6.643856 | -4.605170 |

0.1 | -1 | -3.321928 | -2.302585 |

1 | 0 | 0 | 0 |

2 | 0.301030 | 1 | 0.693147 |

3 | 0.477121 | 1.584963 | 1.098612 |

4 | 0.602060 | 2 | 1.386294 |

5 | 0.698970 | 2.321928 | 1.609438 |

6 | 0.778151 | 2.584963 | 1.791759 |

7 | 0.845098 | 2.807355 | 1.945910 |

8 | 0.903090 | 3 | 2.079442 |

9 | 0.954243 | 3.169925 | 2.197225 |

10 | 1 | 3.321928 | 2.302585 |

20 | 1.301030 | 4.321928 | 2.995732 |

30 | 1.477121 | 4.906891 | 3.401197 |

40 | 1.602060 | 5.321928 | 3.688879 |

50 | 1.698970 | 5.643856 | 3.912023 |

60 | 1.778151 | 5.906991 | 4.094345 |

70 | 1.845098 | 6.129283 | 4.248495 |

80 | 1.903090 | 6.321928 | 4.382027 |

90 | 1.954243 | 6.491853 | 4.499810 |

100 | 2 | 6.643856 | 4.605170 |

200 | 2.301030 | 7.643856 | 5.298317 |

300 | 2.477121 | 8.228819 | 5.703782 |

400 | 2.602060 | 8.643856 | 5.991465 |

500 | 2.698970 | 8.965784 | 6.214608 |

600 | 2.778151 | 9.228819 | 6.396930 |

700 | 2.845098 | 9.451211 | 6.551080 |

800 | 2.903090 | 9.643856 | 6.684612 |

900 | 2.954243 | 9.813781 | 6.802395 |

1000 | 3 | 9.965784 | 6.907755 |

10000 | 4 | 13.287712 | 9.210340 |