Properties of Logarithms
2026-02-28 13:35 Diff
https://brightchamps.com/en-us/math/algebra/properties-of-logarithms

Each property of logarithms aims to simplify and solve logarithmic equations and expressions.


For all logarithmic properties: (m, n > 0, a > 0, a ≠ 1)
Only positive real numbers can be used to define logarithms, and their bases must be positive and not equal to 1. While more properties exist, the four basic properties are listed below: 
 

Product property: 


\( log_a mn = log_a m + log_a n \)
 

Condition: \((m, n > 0, a > 0, a ≠ 1)\)
 

This property states that the logarithm of a product \((mn)\) is equal to the sum of the logarithms of the individual factors. 

Quotient property:


 \(log_a m / n = log_a m - log_a n \)
 

Condition: \((m, n > 0, a > 0, a ≠ 1)\)
 

The quotient property states that the logarithm of a quotient (m/n) is the same as the difference between the logarithms of numerator and denominator.

Power property:


\(log_a(m^n) = n log_a(m)\)
 

Condition: \((m, n > 0, a > 0, a ≠ 1)\)
 

The rule shows that if we move the exponent inside a logarithm to the front of a log, then the exponent will be the multiplier. 

Change of base property:


 \(log_b a = log_c a / log_c b \)
 

Condition: \((m, n > 0, a > 0, a ≠ 1)\)
 

This property states that we can convert the base of a logarithm to another base.  

Other properties are also directly derived from exponent rules. The definition of logarithm is: 

\(a^x = m\) ⇔ \(log_a m = x\)

\(a^0 = 1\) ⇒ \(log_a 1 = 0\) 
 

\(a^1 = a\) ⇒ \(log_a a = 1\)
 

\(a^{\log_a x} = x \)
 

\(\log_b (a^m) = m \cdot \log_b (a) \)

(Results from changing the power property and base rule). 

The following table lists the properties of logarithms.