The Main Idea

• Basic Domain Rule:
  • For [latex]y = \log_b(x)[/latex], the domain is [latex](0, \infty)[/latex]
  • The argument of a logarithm must be positive
• Vertical Asymptote:
  • Logarithmic functions have a vertical asymptote at [latex]x = 0[/latex]
• Inverse Relationship:
  • Domain of [latex]\log_b(x)[/latex] is the range of [latex]b^x[/latex]
  • Range of [latex]\log_b(x)[/latex] is the domain of [latex]b^x[/latex]
• Transformations:
  • Can change the domain of the parent function
  • Always ensure the argument remains positive
• Finding Domains:
  • Set up inequality: [latex]\text{ argument }> 0[/latex]
  • Solve for [latex]x[/latex]
  • Express domain in interval notation

The Main Idea

• Parent Function:
  • [latex]f(x) = \log_b(x)[/latex] where [latex]b > 0[/latex] and [latex]b \neq 1[/latex]
• Inverse Relationship:
  • Logarithmic functions are inverses of exponential functions
  • Their graphs are reflections of each other across [latex]y = x[/latex]
• Key Characteristics:
  • Domain: [latex](0, \infty)[/latex]
  • Range: [latex](-\infty, \infty)[/latex]
  • Vertical asymptote: [latex]x = 0[/latex]
  • [latex]x[/latex]-intercept: [latex](1, 0)[/latex]
  • Key point: [latex](b, 1)[/latex]
• Behavior:
  • Increasing if [latex]b > 1[/latex]
  • Decreasing if [latex]0 < b < 1[/latex]
• Graph Shape:
  • Starts at vertical asymptote [latex]x = 0[/latex]
  • Passes through [latex](1, 0)[/latex] and [latex](b, 1)[/latex]
  • Curves upward ([latex]b > 1[/latex]) or downward ([latex]0 < b < 1[/latex])

The Main Idea

• Parent Function:
  • [latex]f(x) = \log_b(x)[/latex] where [latex]b > 0[/latex] and [latex]b \neq 1[/latex]
• Horizontal Shifts:
  • [latex]f(x) = \log_b(x + c)[/latex]
  • Shifts left [latex]c[/latex] units if [latex]c > 0[/latex]
  • Shifts right [latex]c[/latex] units if [latex]c < 0[/latex]
  • Vertical asymptote: [latex]x = -c[/latex]
  • Domain: [latex](-c, \infty)[/latex]
  • Range: [latex](-\infty, \infty)[/latex]
• Vertical Shifts:
  • [latex]f(x) = \log_b(x) + d[/latex]
  • Shifts up [latex]d[/latex] units if [latex]d > 0[/latex]
  • Shifts down [latex]d[/latex] units if [latex]d < 0[/latex]
  • Vertical asymptote: [latex]x = 0[/latex]
  • Domain: [latex](0, \infty)[/latex]
  • Range: [latex](-\infty, \infty)[/latex]
• Effect on Domain and Range:
  • Horizontal shifts affect domain
  • Vertical shifts do not affect domain or range
• Vertical Stretches and Compressions:
  • [latex]f(x) = a\log_b(x)[/latex], where [latex]a > 0[/latex]
  • Stretch if [latex]a > 1[/latex], compress if [latex]0 < a < 1[/latex]
  • Vertical asymptote: [latex]x = 0[/latex] (unchanged)
  • Domain: [latex](0, \infty)[/latex] (unchanged)
  • Range: [latex](-\infty, \infty)[/latex] (unchanged)
• Reflections:
  • About [latex]x[/latex]-axis: [latex]f(x) = -\log_b(x)[/latex]
    • Domain: [latex](0, \infty)[/latex], Range: [latex](-\infty, \infty)[/latex] (unchanged)
  • About [latex]y[/latex]-axis: [latex]f(x) = \log_b(-x)[/latex]
    • Domain: [latex](-\infty, 0)[/latex], Range: [latex](-\infty, \infty)[/latex]
• Combined Transformations:
  • Can involve multiple operations (e.g., [latex]f(x) = a\log_b(x-h) + k[/latex])
  • Apply transformations in the correct order: inside parentheses first, then outside
• Vertical Asymptote:
  • [latex]x = 0[/latex] for parent function
  • Shifts with horizontal transformations