Logarithms

Definition of Logarithm

A logarithm is the exponent to which a base must be raised to obtain a given number. Logarithms are fundamental in mathematics, especially in calculus, for solving equations involving exponential growth and decay.

• Definition: If , , and , then the logarithm of to base is if and only if . This is written as .

• Notation: is read as "logarithm of to base ".

Example: means . Since , .

Conditions for Logarithms

• Base (): Must be positive and not equal to 1 (, ).

• Argument (): Must be positive ().

Important Properties of Logarithms

Logarithms have several key properties that simplify calculations and solve equations.

• Product Rule:

• Quotient Rule:

• Power Rule:

• Logarithm of 1: (since )

• Logarithm of the Base: (since )

Examples of Logarithm Calculations

• Example 1:

• Example 2 (Change of Base): Additional info: This formula allows you to compute logarithms in any base using a calculator (usually base 10 or base e).

Observations about Logarithms

• Logarithms do not have units.

• There is no logarithm for negative numbers or zero.

• The base is written as a subscript (index) of the log.

Exercises (Sample Problems)

• Calculate the following logarithms:

• Apply properties and simplify:

• Change of base (to base 10):

• Solve for :

Summary Table: Logarithm Properties

Additional info: Logarithms are essential for calculus topics such as derivatives and integrals of exponential and logarithmic functions, and for solving equations involving exponential growth and decay.