Logarithm Rules and Identities

In mathematics, the logarithm is the inverse function to exponentiation. In other words, if we take a logarithm of a quantity, we undo an exponentiation.

For example, if g(x) is a logarithmic function, which could be expressed as

g(x) = y = \log_b x

its inverse function could be expressed in the exponential form

f (x) = b^x

or

x = b^y = b^{\log_{b}x } = x

The conversion involves the use of logarithmic identity m =a^{\log_{a}m }, which we will derive step by step in the article.


In logarithm calculation, we often use a number of rules to rewrite expressions in a variety of different ways. These rules apply to logarithms of any base but the same base must be used throughout a calculation. Since the logarithm is the inverse function to exponentiation, we can derive these basic rules from the rules for exponents.

Name Rules or identities
Product Rule \log_{b}(m\cdot n)=\log_{b}m+\log_{b}n
Quotient Rule \log_{b}(\dfrac{m}{n} ) = \log_{b}m-\log_{b}n
Power Rule \log_{b}(m^n) = n\cdot \log_{b}m
Base Power Rule \log_{b^n} m = \dfrac{1}{n}\log_{b}m
Double Power Rule \log_{b^y}m^x = (\dfrac{x}{y})\log_{b}m
Change-of-Base Rule \log_{b}m = \dfrac{\log_{c}m }{\log_{c}b }

Product Rule for Log

The logarithm of a product is equal to the sum of the logarithm of its individual factors. In a more formal mathmatical language, the product rule of logarithm could be expressed as,

\boxed{\forall b \in \mathbb{R_+}, b\ne 1,\forall m,n, \in \mathbb{R_+},\log_{b}(m\cdot n)=\log_{b}m+\log_{b}n }

The derivation of product rule of logarithm

Let p=\log_{b}m, q=\log_{b}q

From the definition of logarithmic function, we get,

b^p=m and b^q=n

b^p\cdot b^q =a^{p+q}=m\cdot n

Apply the definition of logarithm again

\log_{b}(m\cdot n)=p+q=\log_{b}m+\log_{b}n

Quotient Rule for Logarithm

The logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator to the same base. The formal math statement could be,

\boxed{\forall b \in \mathbb{R_+}, b\ne 1,\forall m,n, \in \mathbb{R_+}, \log_{b}(\dfrac{m}{n} ) = \log_{b}m-\log_{b}n }

The derivation of quotient rule of logarithm

\log_{b}(\dfrac{m}{n} ) = \log_{b}m-\log_{b}n

Let p=\log_{b}m, q=\log_{b}q

Then, b^p=m and b^q=n

The Quotient of the two equations is

\dfrac{b^p}{b^q} = b^{p-q} = \dfrac{m}{n}

Apply the definition of logarithm

\log_{b}(\dfrac{m}{n} ) =p-q = \log_{b}m-\log_{b}n


Power Rule of Logarithm

The logarithm of a power is equals to the product of the exponent times the logarithm of the base. The power rule of logarithm could be expressed formally,

\boxed{\forall b \in \mathbb{R_+}, b\ne 1,\forall m,n, \in \mathbb{R_+}, \log_{b}(m^n) = n\cdot \log_{b}m }

The derivation of power rule of logarithm

Let \log_{b}(m^n) = p

Then b^p = m^n \implies b^{\frac{p}{n} } = m

Apply the definition of logarithm

\log_{b}m = \dfrac{p}{n}

Multiply the equation by n and substitute definition of p

n\cdot \log_{b}m = p =\log_{b}(m^n)

Base Power Rule of Logarithm

Logarithm of a quantity to a base in exponential form is equal to the quotient of logarithm of the quantity by the exponent of the base.

\log_{b^n} m = \dfrac{1}{n}\log_{b}m

\boxed{\forall b \in \mathbb{R_+}, b\ne 1, \forall n \in \mathbb{R}, n\ne 0, \forall m, \in \mathbb{R_+},\log_{b^n} m = \dfrac{1}{n}\log_{b}m }

Let

\log_{b^n} m = p

Then

(b^n)^p = b^{np} =m
(1)

Let

\dfrac{1}{n}\log_{b}m = q

Then

b^{nq} = m
(2)

The following equation is obtained from (1) and (2)

a^{np} =a^{nq}

Therefore, p = q and \log_{b^n} m = \dfrac{1}{n}\log_{b}m

Double Power Rule of Logarithm

Logarithm of a quantity in exponential form to a base in exponential form is equal to product of quotient of exponent of quantity by the exponent of base quantity and logarithm of quantity.

\log_{b^y}m^x = (\dfrac{x}{y})\log_{b}m

\boxed{\forall b \in \mathbb{R_+}, b\ne 1, \forall x,y \in \mathbb{R}, y\ne0, \forall m, \in \mathbb{R_+}, \log_{b^y}m^x = (\dfrac{x}{y})\log_{b}m }


Apply both power rule and base power rules

The Change-of-Base Identity of Logarithm

The logarithm of a quantity could be expressed as the quotient of logarithms of the quantity and its base to a different base.

\boxed{\forall b,c \in \mathbb{R_+}, b,c\ne 1,\forall m,n, \in \mathbb{R_+},\log_{b}m = \dfrac{\log_{c}m }{\log_{c}b } }


p=\log_{b}m \implies b^p=m

\log_{c}b^p =\log_{c}m

p\log_{c}b=\log_{c}m

p=\dfrac{\log_{c}m}{\log_{c}b} =\log_{b}m

Other Logarithmic identities

m =ba^{\log_{b}m }=e^{\ln m}=10^{\log_{10}m }

m=\log_{ba}b^m = \ln e^m=\log_{10}10^m

Collected in the board: Logarithm

Steven Zheng Steven Zheng posted 1 month ago

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