What are the Properties of Logarithms

What are the Properties of Logarithms

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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 $k \neq0$
• $log_a⁡x =\log_{a^c}x^c$
• $log_a⁡x =\frac{1}{\log_x⁡a}$

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