Logarithms Lecture Notes

Introduction

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

Evaluating Logs

• Basic Concept:

  • Logarithm log_b(a) = c means b^c = a
  • log_b(a) asks: "b to what power equals a?"

• Examples:

  • log_2(4) = 2 because 2^2 = 4
  • log_2(8) = 3 because 2^3 = 8
  • log_3(9) = 2 because 3^2 = 9
  • log_4(16) = 2 because 4^2 = 16
  • log_3(27) = 3 because 3^3 = 27
  • log_2(32) = 5 because 2^5 = 32

• Evaluation with Common Bases:

  • If no base is specified (e.g., log(100)), assume base 10
  • log(10) = 1 because 10^1 = 10
  • log(100) = 2 because 10^2 = 100
  • log(1000) = 3 because 10^3 = 1000
  • Count zeros for quick evaluation: log(1,000,000) = 6

• Negative Logs:

  • log(0.1) = -1 because 10^-1 = 0.1
  • log(0.01) = -2 because 10^-2 = 0.01
  • Counting zeros after the decimal: log(0.00001) = -5

Change of Base Formula

• Formula: log_a(b) = log_c(b) / log_c(a)

  • Allows conversion to any base
  • Often convert to base 10 (common logarithm) or base e (natural logarithm)

• Examples:

  • log_2(8) = 3 translates to log(8) / log(2) = 3
  • Convert log_5(x - 1) to natural log: ln(x - 1) / ln(5)

Properties of Logs

• Product Rule: log(a) + log(b) = log(a*b)

• Quotient Rule: log(a) - log(b) = log(a/b)

• Power Rule: log(a^c) = c*log(a)

• Examples - Condensing Logs:

  • log(x) + log(y) - log(z) = log(xy/z)
  • Bring coefficients to exponent positions first, then combine:
    • 2log(x) + 3log(y) - log(z) becomes log(x^2) + log(y^3) - log(z) = log(x^2 * y^3 / z)

Expanding Logs

• Reverse of Condensing:
  • Expand expressions using rules
• Examples - Expanding:
  • log(R^2 * S^5 / Z^6) = 2log(R) + 5log(S) - 6log(Z)
  • Convert radical expressions into exponent form first:
    • log(√(x^2 * y) / z^4) expands to log(x^2 * y^1/2 / z^4) and further to 2log(x) + 0.5log(y) - 4log(z)

Simplifying Log Expressions

• Special Values:

  • Ln(1) = 0
  • Ln(e) = 1
  • Ln(e^5) = 5*Ln(e) = 5

• Simplification Examples: (Using cancellation rules)

  • e^(ln(7)) simplifies to 7
  • 5^(log_5(y^8)) simplifies to y^8
  • log_3(6) - log_3(2) = log_3(3) = 1

Solving Logarithmic Equations

• Basic Conversion: Convert log equations into exponent form to solve

  • log_x(27) = 3 becomes x^3 = 27 ÷ solve x = 3
  • log_2(y) = 5 becomes 2^5 = y ÷ solve y = 32

• Example: Solve log_2(x+5) = 3

  • Convert: 2^3 = x + 5
  • Solve: 8 = x + 5
  • Result: x = 3

• Combining Logs: Apply log rules first

  • log_2(x-3) + log_2(x-1) = 3 becomes log_2((x-3)(x-1)) = 3
  • Solve: Convert to exponent form: 2^3 = (x-3)(x-1) and solve the quadratic equation
  • Check for extraneous solutions: Log of negative numbers doesn't exist

Graphing Logarithmic Functions

• Steps:

  1. Set the inside of the log equal to key values (0, 1, base)
  2. Solve for x to find vertical asymptote and key points.

• Example: Graph y = log_2(x-3) + 1

  • Vertical Asymptote: Set x-3=0 ÷ x=3
  • Table values: Solve for x-3=1, x-3=2 ÷ Points: (4,1) and (5,2)
  • Sketch graph starting from vertical asymptote, passing through points.

• Domain & Range:

  • Exponential Functions:
    • Domain: all real numbers
    • Range: bounded by horizontal asymptote
  • Logarithmic Functions:
    • Domain: starts from vertical asymptote to +∞
    • Range: always -∞ to +∞