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^{5} = 32

Then

log_{2}(32) = 5

### 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

### 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→∞ |

**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) ,

### 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 = 9x

Or

8x = 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 |

Algebra: See Also |