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^{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:
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}(∞) = ∞,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_{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)
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, teh base two logarithm of one is zero:
log_{2}(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_{2}(2) = 1
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_{2}(x) dx = x ∙ ( log_{2}(x) 1 / ln(2)) + C
Natural logarithm is the logarithm to the base e of a number.
Definition of natural logarithm
When
e^{ y} = x
Then base e logarithm of x is
ln(x) = log_{e}(x) = y
The e constant or Euler’s number is:
e ≈ 2.71828183
Ln as inverse function of exponential function
The natural logarithm function ln(x) is the inverse function of the exponential function e^{x}.
For x>0,
f (f ^{1}(x)) = e^{ln(x)} = x
Or
f ^{1}(f (x)) = ln(e^{x}) = x
Natural logarithm rules and properties
Rule name  Rule  Example 

Product rule 
ln(x ∙ y) = ln(x) + ln(y) 
ln(3 ∙ 7) = ln(3) + ln(7) 
Quotient rule 
ln(x / y) = ln(x) – ln(y) 
ln(3 / 7) = ln(3) – ln(7) 
Power rule 
ln(x ^{y}) = y ∙ ln(x) 
ln(2^{8}) = 8∙ ln(2) 
ln derivative 
f (x) = ln(x)⇒ f ‘ (x) = 1 / x 

ln integral 
∫ln(x)dx = x ∙ (ln(x) – 1) + C  
ln of negative number 
ln(x) is undefined when x ≤ 0 

ln of zero 
ln(0) is undefined 

ln of one 
ln(1) = 0  
ln of infinity 
lim ln(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_{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)
Derivative of natural logarithm
The derivative of the natural logarithm function is the reciprocal function.
When
f (x) = ln(x)
The derivative of f(x) is:
f ‘ (x) = 1 / x
Integral of natural logarithm
The integral of the natural logarithm function is given by:
When
f (x) = ln(x)
The integral of f(x) is:
∫ f (x)dx = ∫ln(x)dx = x ∙ (ln(x) – 1) + C
Ln of 0
The natural logarithm of zero is undefined:
ln(0) is undefined
The limit near 0 of the natural logarithm of x, when x approaches zero, is minus infinity:
Ln of 1
The natural logarithm of one is zero:
ln(1) = 0
Ln of infinity
The limit of natural logarithm of infinity, when x approaches infinity is equal to infinity:
lim ln(x) = ∞, when x→∞