Logarithmic Functions
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Domain and Range of Logarithmic Functions
Parent Function:
\[ f(x) = \log_b x \quad \text{where } b > 0, b \neq 1 \]
Domain:
\[ (0, \infty) \quad \text{or} \quad x > 0 \]
Only positive real numbers (the argument must be positive)
Range:
\[ (-\infty, \infty) \quad \text{or all real numbers} \]
The output can be any real number
Transformed Functions:
\[ f(x) = a\log_b(x - h) + k \]
Finding Domain:
1. Set the argument greater than zero: \( x - h > 0 \)
2. Solve for x: \( x > h \)
3. Domain: \( (h, \infty) \)
Range:
Always all real numbers: \( (-\infty, \infty) \)
(Vertical shifts don't affect the range)
Examples:
Find domain and range of: \( f(x) = \log_2(x + 3) \)
Set argument > 0: \( x + 3 > 0 \)
Solve: \( x > -3 \)
Domain: \( (-3, \infty) \)
Range: \( (-\infty, \infty) \)
Find domain and range of: \( g(x) = \log(2x - 6) + 4 \)
Set argument > 0: \( 2x - 6 > 0 \)
Solve: \( 2x > 6 \) → \( x > 3 \)
Domain: \( (3, \infty) \)
Range: \( (-\infty, \infty) \)
Find domain and range of: \( h(x) = \ln(5 - x) \)
Set argument > 0: \( 5 - x > 0 \)
Solve: \( 5 > x \) or \( x < 5 \)
Domain: \( (-\infty, 5) \)
Range: \( (-\infty, \infty) \)
Key Points:
• The domain is determined by the argument being positive
• The range is always all real numbers (unchanged by transformations)
• Horizontal shifts affect the domain
• \( \log_b 0 \) and \( \log_b(\text{negative}) \) are undefined
2. Graph Logarithmic Functions
Parent Function \( f(x) = \log_b x \):
Key Features:
1. Vertical Asymptote:
\( x = 0 \) (y-axis) - the graph approaches but never touches
2. X-intercept:
Always at \( (1, 0) \) because \( \log_b 1 = 0 \)
3. Y-intercept:
NONE (undefined at x = 0)
4. Key Point:
\( (b, 1) \) because \( \log_b b = 1 \)
5. Direction:
• If \( b > 1 \): Increasing (rises from left to right)
• If \( 0 < b < 1 \): Decreasing (falls from left to right)
6. End Behavior:
• As \( x \to 0^+ \): \( f(x) \to -\infty \)
• As \( x \to \infty \): \( f(x) \to \infty \)
Common Key Points for Parent Function:
| x | \( f(x) = \log_2 x \) | \( g(x) = \log_{10} x \) |
|---|---|---|
| 1/4 | -2 | -0.602 |
| 1/2 | -1 | -0.301 |
| 1 | 0 | 0 |
| 2 | 1 | 0.301 |
| 4 | 2 | 0.602 |
| 8 | 3 | 0.903 |
Steps to Graph Parent Function:
1. Draw the vertical asymptote at \( x = 0 \)
2. Plot the x-intercept at \( (1, 0) \)
3. Plot the key point \( (b, 1) \)
4. Plot additional points as needed
5. Draw a smooth curve through points, approaching the asymptote
3. Transformations of Logarithmic Functions
General Form:
\[ f(x) = a\log_b(x - h) + k \]
Parameters:
• a = vertical stretch/compression and reflection
• h = horizontal shift (moves asymptote)
• k = vertical shift
• b = base of logarithm
Horizontal Shifts:
\[ f(x) = \log_b(x - h) \]
• If \( h > 0 \): Shift RIGHT h units
• If \( h < 0 \): Shift LEFT |h| units
• Vertical asymptote moves to \( x = h \)
• X-intercept moves to \( (1 + h, 0) \)
Vertical Shifts:
\[ f(x) = \log_b x + k \]
• If \( k > 0 \): Shift UP k units
• If \( k < 0 \): Shift DOWN |k| units
• Vertical asymptote stays at \( x = 0 \)
• X-intercept changes (solve \( \log_b x + k = 0 \))
Vertical Stretch/Compression and Reflection:
\[ f(x) = a\log_b x \]
• If \( |a| > 1 \): Vertical stretch (steeper)
• If \( 0 < |a| < 1 \): Vertical compression (flatter)
• If \( a < 0 \): Reflection over x-axis
• Vertical asymptote and x-intercept unchanged
Transformation Summary Table:
| Transformation | Effect on Asymptote | Effect on Domain |
|---|---|---|
| \( \log_b(x - h) \) | Moves to \( x = h \) | Becomes \( (h, \infty) \) |
| \( \log_b x + k \) | Unchanged (\( x = 0 \)) | Unchanged \( (0, \infty) \) |
| \( a\log_b x \) | Unchanged (\( x = 0 \)) | Unchanged \( (0, \infty) \) |
4. Graphing Transformed Functions - Examples
Example 1: Horizontal Shift
Graph: \( f(x) = \log_2(x - 3) \)
Transformations:
• Shift right 3 units
Key Features:
• Vertical asymptote: \( x = 3 \)
• X-intercept: \( (4, 0) \) (since \( \log_2(4-3) = \log_2 1 = 0 \))
• Domain: \( (3, \infty) \)
• Range: \( (-\infty, \infty) \)
• Point: \( (5, 1) \) (since \( \log_2 2 = 1 \))
Example 2: Vertical Shift
Graph: \( g(x) = \log x - 2 \)
Transformations:
• Shift down 2 units
Key Features:
• Vertical asymptote: \( x = 0 \)
• X-intercept: \( (100, 0) \) (since \( \log 100 - 2 = 2 - 2 = 0 \))
• Domain: \( (0, \infty) \)
• Range: \( (-\infty, \infty) \)
• Point: \( (1, -2) \), \( (10, -1) \)
Example 3: Combined Transformations
Graph: \( h(x) = -\log_3(x + 1) + 2 \)
Transformations:
• Shift left 1 unit
• Reflect over x-axis (negative sign)
• Shift up 2 units
Key Features:
• Vertical asymptote: \( x = -1 \)
• X-intercept: \( (8, 0) \) (solve \( -\log_3(x+1) + 2 = 0 \))
• Domain: \( (-1, \infty) \)
• Range: \( (-\infty, \infty) \)
• Decreasing function (due to reflection)
5. Relationship with Exponential Functions
Key Relationship:
Logarithmic and exponential functions are inverses of each other
\[ y = \log_b x \quad \Leftrightarrow \quad x = b^y \]
Comparing Graphs:
| Feature | Exponential \( y = b^x \) | Logarithmic \( y = \log_b x \) |
|---|---|---|
| Domain | All reals | \( (0, \infty) \) |
| Range | \( (0, \infty) \) | All reals |
| Asymptote | Horizontal: \( y = 0 \) | Vertical: \( x = 0 \) |
| Intercept | Y-intercept: \( (0, 1) \) | X-intercept: \( (1, 0) \) |
| Key Point | \( (1, b) \) | \( (b, 1) \) |
6. Quick Reference Summary
Essential Features:
Parent Function: \( f(x) = \log_b x \)
Domain: \( (0, \infty) \) (always positive x-values)
Range: \( (-\infty, \infty) \) (all real numbers)
Vertical Asymptote: \( x = 0 \) (or \( x = h \) if shifted)
X-intercept: \( (1, 0) \) for parent function
Key Point: \( (b, 1) \)
Direction: Increasing if \( b > 1 \); Decreasing if \( 0 < b < 1 \)
📚 Study Tips
✓ Domain is always determined by setting argument > 0
✓ Range is always all real numbers regardless of transformations
✓ Horizontal shifts move the vertical asymptote
✓ Logarithmic graphs are reflections of exponential graphs over y = x
✓ Always draw the asymptote first when graphing