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→∞ |

**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

*+ log*

_{b}(x)

_{b}(y)For example:

log_{8}(4 ∙ 2) = log_{8}(4) + log_{8}(2)

**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

*log*

_{b}(x) –

_{b}(y)For example:

log_{8}(5 / 4) = log_{8}(5) – log_{8}(4)

**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_{8}(3^{5}) = 5 ∙ log_{8}(3)

**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_{3}(4) = 1 / log_{4}(3)

**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

*/ log*

_{c}(x)

_{c}(b)For example:

log_{3}(4) = log_{8}(4) / log_{8}(3)

**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:

log_{4}(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_{4}(16) = 2

**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_{4}(x) dx = x ∙ ( log_{4}(x) – 1 / ln(4) ) + C

### Logarithm approximation

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

Algebra: See Also |