Logarithmic functions are the inverse of exponential functions, with unique properties that set them apart. They have a domain of all positive real numbers and a range spanning all real numbers, making them versatile in various applications.

Graphing logarithmic functions involves understanding their key features, such as vertical asymptotes and x-intercepts. Transformations like shifts, stretches, and reflections allow us to manipulate these graphs, providing insights into their behavior and relationships.

Graphs of Logarithmic Functions

Domain and range of logarithms

• Domain of a logarithmic function $f(x) = \log_b(x)$ includes all positive real numbers $(0, \infty)$
  • Argument of the logarithm (expression inside the $\log$) must be greater than 0
  • Logarithms undefined for non-positive numbers (0 and negative numbers)
• Range of a logarithmic function encompasses all real numbers $(-\infty, \infty)$
  • As $x$ approaches 0 from the right, $\log_b(x)$ approaches negative infinity
  • As $x$ approaches positive infinity, $\log_b(x)$ approaches positive infinity

Graphing logarithmic transformations

• Parent logarithmic function $f(x) = \log_b(x)$, where $b$ is the base and $b > 0, b \neq 1$
  • Most common bases are 10 (common log) and $e$ (natural log, ln)
• Logarithmic functions are the inverse of exponential functions
  • If $y = \log_b(x)$, then $b^y = x$
  • This inverse relationship results in a reflection of the exponential function over the line $y = x$
• Transformations of logarithmic functions follow similar rules to other functions
  • Vertical shift: $f(x) = \log_b(x) + k$ shifts the graph up by $k$ units
  • Horizontal shift: $f(x) = \log_b(x - h)$ shifts the graph right by $h$ units
  • Vertical stretch/compression: $f(x) = a \cdot \log_b(x)$
    • Stretches graph vertically by a factor of $|a|$ if $|a| > 1$
    • Compresses graph vertically by a factor of $|a|$ if $0 < |a| < 1$
  • Reflection: $f(x) = -\log_b(x)$ reflects the graph over the $x$-axis

Key features of logarithmic graphs

• Logarithmic functions have a vertical asymptote at $x = 0$
  • Graph approaches the vertical asymptote but never touches or crosses it
• $x$-intercept of a logarithmic function occurs when $y = 0$
  • For the parent function $f(x) = \log_b(x)$, the $x$-intercept is at $(1, 0)$
  • Transformations can shift the $x$-intercept
• Logarithmic functions do not have a $y$-intercept, as $\log_b(0)$ is undefined
  • The $y$-axis serves as the vertical asymptote for the parent function
• End behavior of logarithmic functions:
  1. As $x \to 0^+$, $f(x) \to -\infty$ approaches negative infinity
  2. As $x \to +\infty$, $f(x) \to +\infty$ approaches positive infinity