# Logarithm

## 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

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:

logb(x) = y

For example when:

24 = 16

Then

log2(16) = 4

## Logarithm as inverse function of exponential function

The logarithmic function,

y = logb(x)

is the inverse function of the exponential function,

x = by

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

f (f -1(x)) = blogb(x) = x

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

f -1(f (x)) = logb(bx) = x

## Natural logarithm (ln)

Natural logarithm is a logarithm to the base e:

ln(x) = loge(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) = logb(x)

## Logarithm rules

Rule name Rule

#### Logarithm product rule

logb(x ∙ y) = logb(x) + logb(y)

#### Logarithm quotient rule

logb(x / y) = logb(x) - logb(y)

#### Logarithm power rule

logb(x y) = y ∙ logb(x)

#### Logarithm base switch rule

logb(c) = 1 / logc(b)

#### Logarithm base change rule

logb(x) = logc(x) / logc(b)

#### Derivative of logarithm

f (x) = logb(x) f ' (x) = 1 / ( x ln(b) )

#### Integral of logarithm

logb(x) dx = x ∙ ( logb(x) - 1 / ln(b) ) + C

#### Logarithm of negative number

logb(x) is undefined when x≤ 0

#### Logarithm of 0

logb(0) is undefined

logb(1) = 0

logb(b) = 1

#### Logarithm of infinity

lim logb(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.

logb(x ∙ y) = logb(x) + logb(y)

#### For example:

log10(37) = log10(3) + log10(7)

#### Logarithm quotient rule

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

logb(x / y) = logb(x) - logb(y)

#### For example:

log10(3 / 7) = log10(3) - log10(7)

#### Logarithm power rule

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

logb(x y) = y ∙ logb(x)

#### For example:

log10(28) = 8log10(2)

#### Logarithm base switch rule

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

logb(c) = 1 / logc(b)

#### For example:

log2(8) = 1 / log8(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.

logb(x) = logc(x) / logc(b)

For example, in order to calculate log2(8) in calculator, we need to change the base to 10:

log2(8) = log10(8) / log10(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:

logb(x) is undefined when x ≤ 0

#### Logarithm of 0

The base b logarithm of zero is undefined:

logb(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:

logb(1) = 0

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

log2(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 logb(x) = ∞, when x→∞

See: log of infinity

#### Logarithm of the base

The base b logarithm of b is one:

logb(b) = 1

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

log2(2) = 1

#### Logarithm derivative

When

f (x) = logb(x)

Then the derivative of f(x):

f ' (x) = 1 / ( x ln(b) )

See: log derivative

#### Logarithm integral

The integral of logarithm of x:

logb(x) dx = x ∙ ( logb(x) - 1 / ln(b) ) + C

#### For example:

log2(x) dx = x ∙ ( log2(x) - 1 / ln(2) ) + C

## Logarithm approximation

log2(x) ≈ n + (x/2n - 1) ,

## Complex logarithm

For complex number z:

z = re = x + iy

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

Log z = ln(r) + i(θ+2nπ) = ln(√(x2+y2)) + i·arctan(y/x))

## Logarithm problems and answers

#### Problem #1

Find x for

log2(x) + log2(x-3) = 2

#### Solution:

Using the product rule:

log2(x∙(x-3)) = 2

Changing the logarithm form according to the logarithm definition:

x∙(x-3) = 22

Or

x2-3x-4 = 0

Solving the quadratic equation:

x1,2 = [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

log3(x+2) - log3(x) = 2

#### Solution:

Using the quotient rule:

log3((x+2) / x) = 2

Changing the logarithm form according to the logarithm definition:

(x+2)/x = 32

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 log10x log2x logex
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

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