## Logarithm Rules

The

baseblogarithmof a number is theexponentthat we need to raise thebasein order to get the number.

- Logarithm definition
- Logarithm rules
- Logarithm problems
- Graph of log(x)
- Logarithm table
- Logarithm calculator
## Logarithm definition

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

b=^{ y}xThen the base b logarithm of x is equal to y:

log

(_{b}x)= yFor example when:

2

^{4}= 16Then

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)}=xOr 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:

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 / (xln(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 whenx≤ 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}(∞) =∞,whenx→∞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 whenx≤ 0## Logarithm of 0

The base b logarithm of zero is undefined:

log

_{b}(0) is undefinedThe 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) = 0For example, teh base two logarithm of one is zero:

log

_{2}(1) = 0See: 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) = ∞, whenx→∞See: log of infinity

## Logarithm of the base

The base b logarithm of b is one:

log

_{b}(b) = 1For 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 / (xln(b) )See: log derivative

## Logarithm integral

The integral of logarithm of x:

∫

log(_{b}x)dx=x ∙( log(_{b}x)- 1 / ln(b)) +CFor example:

∫

log_{2}(x)dx=x ∙( log_{2}(x)- 1 / ln(2)) +C## Logarithm approximation

log

_{2}(x) ≈n+ (x/2^{n}- 1) ,## 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)) = 2Changing the logarithm form according to the logarithm definition:

x∙(x-3) = 2^{2}Or

x^{2}-3x-4 = 0Solving the quadratic equation:

x_{1,2}= [3±√(9+16) ] / 2 = [3±5] / 2 = 4,-1Since 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) = 2Changing the logarithm form according to the logarithm definition:

(

x+2)/x= 3^{2}Or

x+2 = 9xOr

8

x= 2Or

x= 0.25## Graph of log(x)

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

## Logarithms table

xlog _{10}xlog _{2}xlog _{e}x0 undefined undefined undefined 0 ^{+}- ∞ - ∞ - ∞ 0.0001 -4.000000 -13.287712 -9.210340 0.001 -3.000000 -9.965784 -6.907755 0.01 -2.000000 -6.643856 -4.605170 0.1 -1.000000 -3.321928 -2.302585 1 0.000000 0.000000 0.000000 2 0.301030 1.000000 0.693147 3 0.477121 1.584963 1.098612 4 0.602060 2.000000 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 3.000000 2.079442 9 0.954243 3.169925 2.197225 10 1.000000 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 2.000000 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 3.000000 9.965784 6.907755 10000 4.000000 13.287712 9.210340

## See also