Lecture on Logarithms

Overview

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

Evaluating Logs

• Logarithms: log base (a) of (b) asks a to what power equals b
• Examples:
  • log₂4 = 2 because 2² = 4
  • log₂8 = 3 because 2³ = 8
  • log₃9 = 2 because 3² = 9
  • You multiply the base; e.g., log base 2 of 32 (2^5 = 32)

Common Logs and Their Properties

• If no base is mentioned, the base is assumed to be 10
• Examples:
  • log10 = 1
  • log100 = 2 (10² = 100)
• Counting zeros with base 10:
  • log1000 = 3, log1000000 = 6 (count zeros)
• Negative quantity inside a log: Not possible
• Log of 1, regardless of base: Always 0

Specific Examples

• log₄16: 4² = 16 so it's 2
• log₅125: 5³ = 125 so it's 3
• If you see a fraction or a smaller base inside the log:
  • log₁₆(1/4) involves negative exponents, hence log₁₆(1/4) = -½
  • 4 raised by fractions without leading to confusion

Change of Base Formula

• logₐb = log_b/logₐ
• You can change any base to another; common choice is natural log (base e)

Expanding and Condensing Logs

• Properties:
  • log(A) + log(B) = log(A*B)
  • log(A) - log(B) = log(A/B)
  • log(A^k) = k*log(A)
• Combining logs involving coefficients and exponents
• Practice Examples:
  • log(X) + log(Y) - log(Z) = log((X*Y)/Z)
  • Always convert coefficients to exponents before combining

Simplifying Logarithmic Expressions

• Key Value Computations:
  • ln(1) = 0, ln(e) = 1
  • e raised to the power of natural logs or simplifications like e^(ln(a)) = a
• Practice Examples
• Use division or multiplication to combine logs if necessary

Solving Logarithmic Equations

• Procedure:
  1. Set logs equal to each other
  2. Convert from log to exponential form if required
  3. Solve for the variable
• Example Problems:
  • log base 3 of 27 = 3 leads to x = 3
  • Consider if solutions are extraneous by substituting back

Graphing Logs and Exponential Functions

• Identify Asymptotes:
  • Logarithmic functions have vertical asymptotes
  • Exponential functions have horizontal asymptotes
• Creating a Table:
  • Set the inside part equal to critical values (0, 1, base)
  • Find corresponding y values
  • Use these points to plot
• Transformations:
  • Reflecting over x or y axis changes the function behavior
  • Symmetry around y=x for inverse functions

Domain and Range Considerations

• Log Functions Domain: Inputs must be greater than 0
• Range: Typically from negative infinity to infinity
• For exponential functions, opposite applies in terms of limitations

Finding the Inverse Functions

• Switch x and y, solve for y
• Use properties of logarithmic and exponential relation
• Check symmetry about y=x

Practice Problems and Applications

• Convert between logarithmic and exponential forms
• Solve real-world problems involving growth and decay

Conclusion

• Mastery involves understanding rules and applying them to different problem types.