The Main Idea 

• Definition of Exponential Functions:
  • Form: [latex]f(x) = ab^x[/latex]
  • [latex]a[/latex] is a non-zero real number (initial value)
  • [latex]b[/latex] is a positive real number, [latex]b \neq 1[/latex] (base)
• Characteristics:
  • Constant base, variable exponent
  • Rapid growth compared to power functions
  • Domain: all real numbers
  • Range: [latex](0, ∞)[/latex] for [latex]b > 1[/latex], [latex](0, ∞)[/latex] for [latex]0 < b < 1[/latex]
• Evaluating Exponential Functions:
  • Substitute the given value for [latex]x[/latex]
  • Follow order of operations
  • Pay attention to parentheses and exponents
• Laws of Exponents:
  • Product of Powers: [latex]b^x \cdot b^y = b^{x+y}[/latex]
  • Quotient of Powers: [latex]\frac{b^x}{b^y} = b^{x-y}[/latex]
  • Power of a Power: [latex](b^x)^y=b^{xy}[/latex]
  • Power of a Product: [latex](ab)^x=a^x b^x[/latex]
  • Power of a Quotient: [latex]\dfrac{a^x}{b^x} =\left(\dfrac{a}{b}\right)^x[/latex]
• Applications:
  • Population growth
  • Compound interest
  • Radioactive decay

The Main Idea 

• Definition of Logarithmic Functions:
  • Inverse of exponential functions
  • [latex]\log_b(x) = y[/latex] if and only if [latex]b^y = x[/latex]
  • Domain: [latex](0, ∞)[/latex], Range: [latex](-∞, ∞)[/latex]
• Common Logarithms:
  • Base 10: [latex]\log_{10}(x)[/latex], often written as [latex]\log(x)[/latex]
  • Natural log: [latex]\log_e(x)[/latex], written as [latex]\ln(x)[/latex]
• Properties of Logarithms:
  • Product: [latex]\log_b(xy) = \log_b(x) + \log_b(y)[/latex]
  • Quotient: [latex]\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)[/latex]
  • Power: [latex]\log_b(x^n) = n\log_b(x)[/latex]
• Change-of-Base Formula:
  • [latex]\log_a(x) = \dfrac{\log_b(x)}{\log_b(a)}[/latex]
  • Useful for calculating logs with non-standard bases
• Solving Logarithmic Equations:
  • Often involves converting to exponential form
  • May require using logarithm properties to simplify

The Main Idea 

• Definitions of Hyperbolic Functions:
  • [latex]\cosh x = \dfrac{e^x + e^{-x}}{2}[/latex]
  • [latex]\sinh x = \dfrac{e^x - e^{-x}}{2}[/latex]
  • [latex]\tanh x = \dfrac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}[/latex]
  • [latex]\text{csch} x = \dfrac{1}{\sinh x}[/latex]
  • [latex]\text{sech} x = \dfrac{1}{\cosh x}[/latex]
  • [latex]\coth x = \dfrac{\cosh x}{\sinh x}[/latex]
• Graphs and Behavior:
  • [latex]\cosh x[/latex]: Similar to [latex]|e^x|[/latex], always [latex]≥ 1[/latex]
  • [latex]\sinh x[/latex]: Odd function, similar to [latex]e^x[/latex] for large x
  • [latex]\tanh x[/latex]: Odd function, approaches [latex]±1[/latex] as [latex]x → ±∞[/latex]
• Key Identities:
  • [latex]\cosh^2 x - \sinh^2 x = 1[/latex]
  • [latex]1 - \tanh^2 x = \text{sech} ^2 x[/latex]
  • [latex]\coth^2 x-1=\text{csch}^2 x[/latex]
  • [latex]\sinh(x \pm y)=\sinh x \cosh y \pm \cosh x \sinh y[/latex]
  • [latex]\cosh (x \pm y)=\cosh x \cosh y \pm \sinh x \sinh y[/latex]
  • [latex]\cosh x + \sinh x = e^x[/latex]
  • [latex]\cosh x-\sinh x=e^{−x}[/latex]
• Inverse Hyperbolic Functions:
  • [latex]\sinh^{-1} x = \ln(x + \sqrt{x^2 + 1})[/latex]
  • [latex]\cosh^{-1} x = \ln(x + \sqrt{x^2 - 1})[/latex] ([latex]x ≥ 1[/latex])
  • [latex]\tanh^{-1} x = \frac{1}{2}\ln(\frac{1+x}{1-x})[/latex] ([latex]|x| < 1[/latex])