Properties of Logarithms

Logarithms are mathematical operations that are the inverse of exponentiation. They have several important properties that make them useful in various mathematical contexts, particularly in solving exponential equations and simplifying expressions. Understanding the properties of logarithms is essential for students preparing for exams, as they often form the basis of more complex problems.

Basic Definition

Before we delve into the properties, let's define what a logarithm is. The logarithm of a number is the exponent to which a specified base must be raised to obtain that number. It is written as:

$$ \log_b(a) = c \quad \text{if and only if} \quad b^c = a $$

where:

• $b$ is the base of the logarithm
• $a$ is the number we are taking the logarithm of (the argument)
• $c$ is the value of the logarithm

Properties of Logarithms

There are several key properties of logarithms that are useful in simplifying expressions and solving equations:

1. Product Property: The logarithm of a product is the sum of the logarithms of the factors.

   $$ \log_b(mn) = \log_b(m) + \log_b(n) $$

2. Quotient Property: The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator.

   $$ \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) $$

3. Power Property: The logarithm of a number raised to an exponent is the exponent times the logarithm of the number.

   $$ \log_b(m^p) = p \cdot \log_b(m) $$

4. Change of Base Formula: The logarithm to one base can be converted to another base by using the following formula.

   $$ \log_b(a) = \frac{\log_k(a)}{\log_k(b)} $$

   where $k$ is a new base and it is often chosen to be 10 or $e$ (the natural logarithm base) for convenience.

5. Logarithm of 1: The logarithm of 1 to any base is 0.

   $$ \log_b(1) = 0 $$

6. Logarithm of the Base: The logarithm of the base to itself is 1.

   $$ \log_b(b) = 1 $$

7. Logarithm of a Negative Number: Logarithms of negative numbers are not defined in the real number system.

   $$ \log_b(-a) \quad \text{is undefined for real numbers} $$

8. Logarithm of Zero: Logarithms of zero are not defined.

   $$ \log_b(0) \quad \text{is undefined} $$

Table of Properties

Property	Formula	Example
Product	$\log_b(mn) = \log_b(m) + \log_b(n)$	$\log_2(8 \cdot 3) = \log_2(8) + \log_2(3)$
Quotient	$\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)$	$\log_5\left(\frac{25}{5}\right) = \log_5(25) - \log_5(5)$
Power	$\log_b(m^p) = p \cdot \log_b(m)$	$\log_3(9^2) = 2 \cdot \log_3(9)$
Change of Base	$\log_b(a) = \frac{\log_k(a)}{\log_k(b)}$	$\log_2(8) = \frac{\log_{10}(8)}{\log_{10}(2)}$
Logarithm of 1	$\log_b(1) = 0$	$\log_7(1) = 0$
Logarithm of the Base	$\log_b(b) = 1$	$\log_4(4) = 1$
Logarithm of a Negative Number	Undefined	$\log_2(-4)$ is undefined
Logarithm of Zero	Undefined	$\log_10(0)$ is undefined

Examples

Example 1: Product Property

Simplify $\log_2(32 \cdot 8)$ using the product property.

$$ \begin{align*} \log_2(32 \cdot 8) &= \log_2(32) + \log_2(8) \ &= 5 + 3 \ &= 8 \end{align*} $$

Example 2: Quotient Property

Simplify $\log_3\left(\frac{81}{3}\right)$ using the quotient property.

$$ \begin{align*} \log_3\left(\frac{81}{3}\right) &= \log_3(81) - \log_3(3) \ &= 4 - 1 \ &= 3 \end{align*} $$

Example 3: Power Property

Simplify $\log_5(125^2)$ using the power property.

$$ \begin{align*} \log_5(125^2) &= 2 \cdot \log_5(125) \ &= 2 \cdot 3 \ &= 6 \end{align*} $$

Example 4: Change of Base Formula

Calculate $\log_2(32)$ using the change of base formula with base 10.

$$ \begin{align*} \log_2(32) &= \frac{\log_{10}(32)}{\log_{10}(2)} \ &= \frac{1.50515}{0.30103} \ &\approx 5 \end{align*} $$

Understanding these properties and being able to apply them to simplify logarithmic expressions is crucial for students preparing for exams. Practice with a variety of problems will help solidify these concepts.