AP Precalculus: Logarithmic Functions Formulas & Graphs
1. Logarithmic Function General Form
\( f(x) = a\log_b(x-h) + k \)
- \( a \) is a stretch/compression/reflection
- \( b \) is the base (\( b>0, b \neq 1 \))
- \( h \) shifts horizontally; \( k \) vertically
2. Domain and Range
- Domain: \( x-h > 0 \) (so \( x > h \))
- Range: all real numbers (\( -\infty < y < \infty \))
- Vertical asymptote: \( x = h \)
3. Graphing Logarithmic Functions
- Basic: \( f(x) = \log_b x \)
- Domain: \( x > 0 \); Range: all reals; Vertical asymptote: \( x = 0 \)
- If \( a > 0 \): Increases right, passes through \( (1,0) \) if no shift
- If \( a < 0 \): Decreases right (reflected in x-axis)
- Horizontal shift: Right by \( h \); graph’s vertical asymptote at \( x = h \)
- Vertical shift: Moves curve up or down by \( k \)