Logarithms: Definition and Properties

Inverse Functions and Logarithms

The logarithm with base a, written as , is defined as the inverse of the exponential function . This means that logarithms 'undo' exponentials, and vice versa.

• Definition: If , then .

• Example: because .

Graphical Properties:

• The domain of is .

• The graph always passes through , since .

• End behavior: as .

• Vertical asymptote at : as .

• Logarithm and exponential with the same base cancel each other: and .

Logarithm Rules

Logarithms have several important algebraic properties:

• Exponents can be brought down:

The Natural Logarithm

Definition and Properties

The natural logarithm is the logarithm with base e, written as or . In Calculus, $\ln(x)$ is preferred because it has the simplest derivative and is closely related to the exponential function .

Change of Base Formula

All logarithms can be converted to natural logarithms using the change of base formula:

Solving Equations with Exponentials and Logarithms

Example 1: Solving Exponential Equations

Problem: Find all solutions to .

• Isolate the exponential:

• Apply natural logarithm:

• (since )

• Solve for :

Example 2: Solving Logarithmic Equations

Problem: Solve for .

• Isolate :

• Exponentiate both sides:

• Final answer:

Simplifying Logarithmic Expressions

Example 3: Simplification

• (a)

  • Note:

• (b)

  • Use change of base:

  • Since , the expression simplifies to

Summary Table: Logarithm Properties

Additional info: The notes emphasize the importance of the natural logarithm in Calculus due to its simple derivative and its equivalence to other logarithms via the change of base formula. All logarithmic manipulations in Calculus can be performed using .