Exponential and logarithmic functions are powerful tools for modeling growth, decay, and complex relationships. They're inverses of each other, with exponentials growing rapidly and logarithms slowing down as x increases.

These functions pop up everywhere, from compound interest to population dynamics. Understanding their properties and rules is key to solving real-world problems and grasping more advanced math concepts.

Exponential Functions

Graphing exponential functions

• General form $f(x) = a \cdot b^x$ represents exponential functions
  • Vertical stretch factor and y-intercept determined by $a$ (0, a)
  • Base of the exponential function given by $b$
    • Function increases (grows) as x increases when $b > 1$ (2, e)
    • Function decreases (decays) as x increases when $0 < b < 1$ (1/2, 1/e)
• Exponential growth modeled by $f(x) = a \cdot (1 + r)^x$ with growth rate $r > 0$ (5%, 0.1)
• Exponential decay represented by $f(x) = a \cdot (1 - r)^x$ with decay rate $0 < r < 1$ (2%, 0.05)
  • Half-life is the time required for a quantity to reduce to half its initial value in exponential decay
• Horizontal asymptote $y = 0$ exists for exponential functions with $0 < b < 1$

Comparison of exponential bases

• Exponential functions with $b > 1$ always increase
  • Steeper growth results from larger bases (2 vs 3, e vs 10)
• Exponential functions with $0 < b < 1$ always decrease
  • Steeper decay caused by smaller bases (1/2 vs 1/3, 1/e vs 0.1)
• Point (0, 1) is common to all exponential functions

Significance of natural base e

• Mathematical constant $e \approx 2.71828$ known as the natural base
• Natural exponential functions $f(x) = e^x$ have base $e$
  • Model continuous growth or decay processes (population, radioactivity)
• Applications of natural exponential functions include:
  • Continuous compound interest (bank accounts, investments)
  • Population growth models (bacteria, viral spread)
  • Radioactive decay (carbon dating, nuclear physics)

Logarithmic Functions

Logarithmic functions and graphs

• General form $f(x) = \log_b(x)$ represents logarithmic functions, where $b > 0$ and $b \neq 1$
  • Logarithm $\log_b(x)$ gives the exponent to which $b$ must be raised to get $x$
  • Common logarithm (base 10) is written as $\log(x)$ or $\log_{10}(x)$
  • Natural logarithm (base e) is denoted as $\ln(x)$ or $\log_e(x)$
• Domain restricted to $x > 0$ as logarithms only defined for positive real numbers
• Range includes all real numbers
• Vertical asymptote occurs at $x = 0$
• Logarithmic functions serve as the inverse of exponential functions

Exponential vs logarithmic functions

• Inverse relationship: If $y = b^x$, then $x = \log_b(y)$
  • Example: If $y = 2^x$, then $x = \log_2(y)$
• Graphically, exponential and logarithmic functions reflect each other over the line $y = x$

Change of base in logarithms

• Change of base formula $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$ allows calculation of logarithms with any base
  • Requires $a > 0$, $a \neq 1$, and $b > 0$, $b \neq 1$
  • Enables use of common bases (10, e) on calculators for logarithms with different bases

Hyperbolic Functions

Properties of hyperbolic functions

• Hyperbolic sine $\sinh(x) = \frac{e^x - e^{-x}}{2}$
  • Odd function with domain and range of all real numbers
• Hyperbolic cosine $\cosh(x) = \frac{e^x + e^{-x}}{2}$
  • Even function with domain of all real numbers and range $y \geq 1$
• Hyperbolic tangent $\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$
  • Odd function with domain of all real numbers and range $-1 < y < 1$
  • Horizontal asymptotes at $y = 1$ and $y = -1$
• Hyperbolic functions analogous to trigonometric functions but defined using exponential functions instead of circular functions

Exponent and Logarithm Rules

Exponent Rules

• Product rule: $a^m \cdot a^n = a^{m+n}$
• Quotient rule: $\frac{a^m}{a^n} = a^{m-n}$
• Power rule: $(a^m)^n = a^{mn}$
• Zero exponent rule: $a^0 = 1$ (for $a \neq 0$)
• Negative exponent rule: $a^{-n} = \frac{1}{a^n}$

Logarithm Rules

• Product rule: $\log_b(xy) = \log_b(x) + \log_b(y)$
• Quotient rule: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
• Power rule: $\log_b(x^n) = n\log_b(x)$
• Change of base: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$

Solving Exponential and Logarithmic Equations

• Exponential equations often require logarithms to solve
• Logarithmic equations may require exponentials to solve
• Use inverse functions and properties to isolate variables