Lecture on Logarithms

Key Topics Covered

• Evaluating logs
• Change of base formula
• Expanding and condensing logs
• Solving logarithmic equations
• Graphing logarithmic functions

Evaluating Logs

• Log base 2 of 4: Ask '2 to what power is 4?' Answer: 2 (since 2^2 = 4).
• Log base 2 of 8: '2 to what power is 8?' Answer: 3 (since 2^3 = 8).
• Log base 3 of 9: '3 to what power is 9?' Answer: 2 (since 3^2 = 9).
• Log base 4 of 16: Answer: 2 (since 4^2 = 16).
• Log base 3 of 27: Answer: 3 (since 3^3 = 27).
• Log base 2 of 32: Answer: 5 (since 2^5 = 32).

More Examples

• Log base 5 of 125: Answer: 3 (since 5^3 = 125).
• Log base 6 of 36: Answer: 2 (since 6^2 = 36).
• Log base 2 of 64: Answer: 6 (since 2^6 = 64).
• Log base 3 of 81: Answer: 4 (since 3^4 = 81).
• Log base 7 of 49: Answer: 2 (since 7^2 = 49).
• Log base 3 of 1: Answer: 0 (since any number to the 0 power is 1).
• Log of 10, 100, 1000, 1 million: Count zeros; answers are 1, 2, 3, 6 respectively.

Logs with Fractions and Negative Numbers

• Log of a fraction (e.g., log 0.1): Negative powers. E.g., log 0.1 = -1, log 0.01 = -2.
• Log of 0 or negative numbers: Does not exist. Logarithms require positive inputs.

Change of Base Formula

• Formula: log base a of b = log b / log a
• Useful for calculating logs with a calculator.

Properties of Logs

• Log Addition: log(a) + log(b) = log(a * b)
• Log Subtraction: log(a) - log(b) = log(a / b)
• Log of a Power: log(a^b) = b * log(a)

Expanding and Condensing Logs

• Condensing: Combine multiple logs into a single log expression.
• Expanding: Break down a single log expression into multiple logs.

Solving Logarithmic Equations

• Convert logs to exponential form to find unknowns.
• Check for extraneous solutions, especially with negative values.

Graphing Logarithms

• Vertical Asymptotes: Located at the value where the log is undefined.
• Domain: Typically starts from the vertical asymptote to infinity.
• Range: Always negative infinity to infinity.

Inverse Functions

• Exponential and Logarithmic Functions: Inverse of each other.
• Graphical Relationship: Reflected across the line y = x.

Practice Problems

• Evaluate logarithms using properties and change of base formula.
• Solve equations by converting between log and exponential forms.
• Graph log functions and their inverses.
• Use sign charts to determine the domain of log functions with multiple critical points.

These notes provide a comprehensive overview of logarithms, their properties, and methods for solving equations and graphing functions. They include practical strategies for evaluating and transforming logarithmic expressions, as well as considerations for graphing and understanding their relationships with exponential functions.