1.5 Exponential and Logarithmic Functions

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

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• 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