Logarithmic functions are powerful tools for modeling growth and decay. They're the inverse of exponential functions, with unique properties like a vertical asymptote at x=0 and an x-intercept at (1,0).

Graphing logarithms involves understanding transformations like shifts and stretches. The domain is all positive real numbers, while the range includes all reals. Key features include continuity, monotonicity, and concave-down behavior for x>0.

Graphing Logarithmic Functions

Graphing logarithmic functions

• The parent function for logarithms $f(x) = \log_b(x)$
  • $b$ represents the base of the logarithm ($b > 0, b \neq 1$)
  • Most common bases are 10 (common log) and $e$ (natural log, ln)
• Transformations applied to graph logarithmic functions
  • Vertical shift $f(x) = \log_b(x) + k$ moves the graph up by $k$ units ($k > 0$) or down by $|k|$ units ($k < 0$)
  • Horizontal shift $f(x) = \log_b(x - h)$ moves the graph right by $h$ units ($h > 0$) or left by $|h|$ units ($h < 0$)
  • Vertical stretch/compression $f(x) = a \cdot \log_b(x)$ stretches the graph vertically by a factor of $|a|$ ($|a| > 1$) or compresses the graph vertically by a factor of $|a|$ ($0 < |a| < 1$)
    • $a < 0$ also reflects the graph over the x-axis
  • Horizontal stretch/compression $f(x) = \log_b(c \cdot x)$ compresses the graph horizontally by a factor of $c$ ($c > 1$) or stretches the graph horizontally by a factor of $\frac{1}{c}$ ($0 < c < 1$)
    • $c < 0$ also reflects the graph over the y-axis (inverse function)

Domain and range of logarithms

• Domain of a logarithmic function includes all positive real numbers, excluding 0
  • Argument of the logarithm (value inside parentheses) must be greater than 0
• Range of a logarithmic function includes all real numbers
  • As $x$ approaches 0 from the right, $\log_b(x)$ approaches negative infinity
  • As $x$ approaches positive infinity, $\log_b(x)$ approaches positive infinity

Key features of logarithmic graphs

• Logarithmic functions have a vertical asymptote at $x = 0$
  • Graph approaches the asymptote but never touches or crosses it
• X-intercept of a logarithmic function occurs when $\log_b(x) = 0$
  • Happens when $x = 1$, so x-intercept is always $(1, 0)$ for the parent function
  • For transformed functions, x-intercept is $(1 + h, 0)$, where $h$ represents the horizontal shift
• Logarithmic functions do not have a y-intercept, as $\log_b(0)$ is undefined
• End behavior
  1. As $x$ approaches 0 from the right, $\log_b(x)$ approaches negative infinity
  2. As $x$ approaches positive infinity, $\log_b(x)$ approaches positive infinity
     • Graph increases slowly for large values of $x$

Additional properties of logarithmic functions

• Continuity: Logarithmic functions are continuous for all $x > 0$
• Monotonicity: Logarithmic functions are strictly increasing (monotonic) for all $x > 0$
• Concavity: Logarithmic functions are always concave down for all $x > 0$