What are the Properties of Logarithms

What are the Properties of Logarithms

 Read,4 minutes

Logarithmic properties

In math, logarithm problems are solved using the properties of the functions that make up logarithms. In basic math, we learned a lot of properties, like commutative, associative, and distributive, that can be used in algebra. When it comes to logarithmic functions, there are five main things to know:

• Quotient Property
• Power rule
• Change of Base Rule
• Reciprocal rule
• Product Property

Product Property

Let \(b, \ m, \ n\) be positive integers and \(b \ ≠ \ 1\), \(log_b \ (mn \ ) \ = \ log_b \ m \ + \ log_b \ n\).
So, the \(log\) of the product of two numbers \(m\) and \(n\) with base \(b\) is the same as adding the \(log\) of \(m\) and \(n\) with the same base \(b\).

Example:

\(log_2 \ (32 \times 16 \ ) \ = \ log_2 \ 32 \ + \ log_2 \ 16 \ = \ log_2 \ 2^5 \ + \ log_2 \ 2^4 \ = \ 5log_2 \ 2 \ + \ 4log_2 \ 2 \ = \ 5 \ + \ 4 \ = \ 9\)

Quotient Property

Let \(b, \ m, \ n\) be positive integers and \(b \ ≠ \ 1\), \(log_b \ (\frac{m}{n}) \ = \ log_b \ m \ - \ log_b \ n\).
The difference between \(log \ m\) and \(log \ n\) with the same base \(b\) is equal to the logarithm of a quotient of \(m\) and \(n\).

Example:

\(log_3 \ (\frac{81}{729}) \ = \ log_3 \ 81 \ - \ log_3 \ 729 \ = \ log_3 \ 3^4 \ - \ log_3 \ 3^6 \ = \ 4log_3 \ 3 \ - \ 6log_3 \ 3 \ = \ 4 \ - \ 6 \ = \ -2\)

Power Rule

Let \(b, \ m, \ n\) be positive integers and \(b \ ≠ \ 1\), \(log_b \ m^n \ = \ nlog_b \ m\).

The above property says that the logarithm of a positive number \(m\) to the power of \(n\) is equal to the product of \(n\) and the \(log\) of \(m\).

Example:

The most important properties of logarithms are the above three. Here are some other properties and examples of how to use them.

Change of Base Rule 

Let \(a, \ b, \ m\) be positive integers and \(a \ ≠ \ 1\), \(b \ ≠ \ 1\), \(log_a \ m \ = \ \frac{log_b \ m}{log_b \ a}\).

Example:

\(log_5 \ 25 \ = \ \frac{log_2 \ 25}{log_2 \ 5}\)

Reciprocal Rule

Let \(m, \ n\) be positive integers and \(m \ ≠ \ 1\), \(n \ ≠ \ 1\), \(log_n \ m \ = \ \frac{1}{log_m \ n}\).

Example:

\(log_3 \ 12 \ = \ \frac{1}{log_{12} \ 3}\)

Summary:

• \({a^{\log_a⁡b}}={b}\)
• \(log_a⁡1 = 0\)
• \(log_a⁡a = 1\)
• \(log_a⁡(x .y) = \log_a⁡x+\log_a⁡y\)
• \(log_a\frac{x}{y}=\log_a⁡x-\log_a⁡y\)
• \(\log_a \frac{1}{x}=-\log_a⁡x\)
• \(log_a⁡ x^p= p\log_a⁡x\)
• \(log_{x^k}⁡x =\frac{1}{x}\log_a⁡x,\)  for \(k \neq0\)
• \(log_a⁡x =\log_{a^c}x^c\)
• \(log_a⁡x =\frac{1}{\log_x⁡a}\)

Free printable Worksheets