The Main Idea

• Logarithmic-Exponential Equivalence:
  • [latex]y = \log_b(x)[/latex] is equivalent to [latex]x = b^y[/latex]
  • Logarithms are the inverse of exponential functions
• Logarithm Definition:
  • For [latex]x > 0, b > 0, b \neq 1[/latex]:
  • [latex]y = \log_b(x)[/latex] means [latex]b^y = x[/latex]
• Domain and Range:
  • Domain of logarithm function: [latex](0, ∞)[/latex]
  • Range of logarithm function: [latex](-∞, ∞)[/latex]
• Base-10 Convention:
  • If no base is specified, assume base 10
• Conversion Process:
  • Exponential to Logarithmic: Identify base, exponent, and result
  • Logarithmic to Exponential: Identify base, argument, and result

The Main Idea

• Mental Evaluation:
  • Use knowledge of squares, cubes, and roots to evaluate logarithms mentally
  • Remember: “base to the exponent gives us the number”
• Logarithm as Exponent:
  • In [latex]\log_b(x) = y[/latex], [latex]y[/latex] is the exponent to which [latex]b[/latex] must be raised to get [latex]x[/latex]
• Inverse Relationship:
  • [latex]\log_b(x) = y[/latex] is equivalent to [latex]b^y = x[/latex]
• Negative Exponents:
  • Remember [latex]b^{-a} = \frac{1}{b^a}[/latex] for evaluating logarithms of fractions
• Complex Logarithms:
  • Even seemingly complicated logarithms can often be evaluated mentally by breaking them down

The Main Idea

• Definition:
  • A common logarithm is a logarithm with base 10
  • Written as [latex]\log(x)[/latex] or [latex]\log_{10}(x)[/latex]
• Relationship:
  • [latex]y = \log(x)[/latex] is equivalent to [latex]10^y = x[/latex]
• Inverse Function:
  • [latex]\log(10^x) = x[/latex] for all [latex]x[/latex]
  • [latex]10^{\log(x)} = x[/latex] for [latex]x > 0[/latex]
• Applications:
  • Richter Scale for earthquakes
  • Stellar magnitude scale for star brightness
  • pH scale for acidity and alkalinity
• Evaluation:
  • Exact values for powers of 10
  • Approximate values using calculator for other numbers

The Main Idea

• Definition:
  • A natural logarithm is a logarithm with base [latex]e[/latex]
  • Written as [latex]\ln(x)[/latex] or [latex]\log_e(x)[/latex]
• Relationship:
  • [latex]y = \ln(x)[/latex] is equivalent to [latex]e^y = x[/latex]
• Inverse Function:
  • [latex]\ln(e^x) = x[/latex] for all [latex]x[/latex]
  • [latex]e^{\ln(x)} = x[/latex] for [latex]x > 0[/latex]
• Key Properties:
  • [latex]\ln(1) = 0[/latex]
  • [latex]\ln(e) = 1[/latex]
• Evaluation:
  • Most values require a calculator
  • Exception: powers of [latex]e[/latex] can be evaluated using inverse property
• Domain:
  • Natural logarithms are only defined for positive real numbers