Logarithmic Functions

📌 What is a Logarithmic Function?

A logarithmic function is the inverse of an exponential function. The parent logarithmic function is:

\( f(x) = \log_b x \)

Where \( b \) is the base (\( b > 0, b \neq 1 \)) and \( x \) is the argument (must be positive)

Domain and Range of Logarithmic Functions

Parent Function: \( f(x) = \log_b x \)

Domain:

\( x > 0 \) or \( (0, \infty) \)

The argument must be positive (you cannot take the log of zero or negative numbers)

Range:

All real numbers or \( (-\infty, \infty) \)

The output can be any real number (positive, negative, or zero)

Transformed Function: \( f(x) = a\log_b(x - h) + k \)

Domain:

Set the argument greater than zero and solve:

\( x - h > 0 \) → \( x > h \) or \( (h, \infty) \)

Range:

All real numbers: \( (-\infty, \infty) \)

Vertical shifts and stretches don't change the range

📝 Examples - Finding Domain:

Example 1: Find domain of \( f(x) = \log_2(x + 3) \)

Set argument > 0: \( x + 3 > 0 \)
Solve: \( x > -3 \)
Domain: \( (-3, \infty) \)

Example 2: Find domain of \( f(x) = \log_5(2x - 4) + 1 \)

Set argument > 0: \( 2x - 4 > 0 \)
Solve: \( 2x > 4 \) → \( x > 2 \)
Domain: \( (2, \infty) \)
Range: All real numbers

Example 3: Find domain of \( f(x) = \ln(5 - x) \)

Set argument > 0: \( 5 - x > 0 \)
Solve: \( -x > -5 \) → \( x < 5 \)
Domain: \( (-\infty, 5) \)

Key Features of Logarithmic Graphs

Characteristics of \( f(x) = \log_b x \) (where \( b > 1 \)):

Vertical Asymptote: \( x = 0 \) (y-axis)
No Horizontal Asymptote
X-intercept: \( (1, 0) \) — because \( \log_b 1 = 0 \)
No Y-intercept — cannot evaluate at \( x = 0 \)
Passes through: \( (b, 1) \) — because \( \log_b b = 1 \)
Passes through: \( \left(\frac{1}{b}, -1\right) \) — because \( \log_b \frac{1}{b} = -1 \)
Always increasing (rises from left to right)
Concave down (growth rate decreases)
Continuous for all \( x > 0 \)

End Behavior:

As \( x \to 0^+ \) (approaches 0 from the right):

\( f(x) \to -\infty \)

The graph goes down toward negative infinity near the asymptote

As \( x \to \infty \):

\( f(x) \to \infty \)

The graph increases slowly but continues to rise

Transformations of Logarithmic Functions

General Form:

\( f(x) = a\log_b(x - h) + k \)

Transformation Effects:

\( a \): Vertical stretch/compression and reflection
- If \( |a| > 1 \): vertical stretch
- If \( 0 < |a| < 1 \): vertical compression
- If \( a < 0 \): reflection over x-axis
\( h \): Horizontal shift
- If \( h > 0 \): shift RIGHT \( h \) units
- If \( h < 0 \): shift LEFT \( |h| \) units
- Asymptote moves to \( x = h \)
\( k \): Vertical shift
- If \( k > 0 \): shift UP \( k \) units
- If \( k < 0 \): shift DOWN \( |k| \) units

Special Transformations:

Reflection over x-axis:

\( f(x) = -\log_b x \)

Reflection over y-axis:

\( f(x) = \log_b(-x) \)

Domain becomes \( (-\infty, 0) \) (negative values only)

Graphing Logarithmic Functions

Step-by-Step Graphing Process:

Identify the parent function \( \log_b x \)
Find the vertical asymptote
- Set argument = 0: for \( \log_b(x-h) \), asymptote is \( x = h \)
Find the x-intercept
- Set \( f(x) = 0 \) and solve
- For parent function: x-intercept is (1, 0)
Plot key points
- Use points like \( (b, 1) \), \( (1, 0) \), \( (\frac{1}{b}, -1) \) for parent
- Apply transformations to these points
Sketch the curve
- Graph approaches asymptote on left
- Graph increases slowly to the right
State domain and range

📝 Example 1 - Parent Function:

Graph \( f(x) = \log_2 x \)

Step 1: Vertical asymptote at \( x = 0 \)

Step 2: X-intercept at \( (1, 0) \)

Step 3: Key points:

\( x = \frac{1}{2} \): \( f(\frac{1}{2}) = \log_2 \frac{1}{2} = -1 \) → \( (\frac{1}{2}, -1) \)
\( x = 1 \): \( f(1) = \log_2 1 = 0 \) → \( (1, 0) \)
\( x = 2 \): \( f(2) = \log_2 2 = 1 \) → \( (2, 1) \)
\( x = 4 \): \( f(4) = \log_2 4 = 2 \) → \( (4, 2) \)
\( x = 8 \): \( f(8) = \log_2 8 = 3 \) → \( (8, 3) \)

Step 4: Domain: \( (0, \infty) \), Range: \( (-\infty, \infty) \)

📝 Example 2 - Horizontal Shift:

Graph \( f(x) = \log_3(x - 2) \)

Step 1: Identify transformation

Shift right 2 units

Step 2: Vertical asymptote

\( x - 2 = 0 \) → \( x = 2 \)

Step 3: X-intercept

Set \( f(x) = 0 \): \( \log_3(x-2) = 0 \)
\( x - 2 = 1 \) → \( x = 3 \)
X-intercept: \( (3, 0) \)

Step 4: Key points (shift parent points right 2)

\( (3, 0) \), \( (5, 1) \), \( (11, 2) \)

Domain: \( (2, \infty) \), Range: \( (-\infty, \infty) \)

📝 Example 3 - Multiple Transformations:

Graph \( f(x) = -2\log_2(x + 1) + 3 \)

Transformations:

Shift LEFT 1 unit: \( (x + 1) \)
Vertical stretch by 2: coefficient 2
Reflection over x-axis: negative sign
Shift UP 3 units: \( +3 \)

Vertical asymptote:

\( x + 1 = 0 \) → \( x = -1 \)

X-intercept:

Set \( f(x) = 0 \): \( -2\log_2(x+1) + 3 = 0 \)
\( -2\log_2(x+1) = -3 \)
\( \log_2(x+1) = \frac{3}{2} \)
\( x + 1 = 2^{3/2} = 2\sqrt{2} \approx 2.83 \)
\( x \approx 1.83 \)

Domain: \( (-1, \infty) \), Range: \( (-\infty, \infty) \)

📝 Example 4 - Natural Logarithm:

Graph \( f(x) = \ln(x - 3) + 2 \)

Vertical asymptote: \( x = 3 \)

X-intercept:

\( \ln(x-3) + 2 = 0 \)
\( \ln(x-3) = -2 \)
\( x - 3 = e^{-2} \)
\( x = 3 + e^{-2} \approx 3.135 \)

Key points:

\( x = 4 \): \( f(4) = \ln(1) + 2 = 0 + 2 = 2 \) → \( (4, 2) \)
\( x = 3 + e \approx 5.72 \): \( f \approx \ln(e) + 2 = 3 \) → \( (5.72, 3) \)

Domain: \( (3, \infty) \), Range: \( (-\infty, \infty) \)

Comparing Logarithmic and Exponential Functions

Inverse Relationship:

Logarithmic functions are inverses of exponential functions, so their graphs are reflections across the line \( y = x \).

Property	Exponential \( b^x \)	Logarithmic \( \log_b x \)
Domain	All real numbers	\( x > 0 \)
Range	\( y > 0 \)	All real numbers
Asymptote	Horizontal: \( y = 0 \)	Vertical: \( x = 0 \)
Intercept	Y-intercept: \( (0, 1) \)	X-intercept: \( (1, 0) \)
Behavior	Rapid growth	Slow growth

⚡ Quick Summary

Domain: Set argument > 0 and solve for \( x \)
Range: Always all real numbers (no restriction)
Vertical asymptote: Where argument = 0
X-intercept: Set function = 0 and solve
No y-intercept (cannot evaluate at \( x = 0 \))
Parent function passes through \( (1, 0) \), \( (b, 1) \), \( (\frac{1}{b}, -1) \)
Always increasing (for \( b > 1 \))
Horizontal shift \( h \) moves asymptote to \( x = h \)
Vertical shifts do not affect domain
Graph is the reflection of exponential across \( y = x \)

📚 Common Mistakes to Avoid

❌ Don't include zero or negative numbers in the domain
❌ Don't forget that \( \log_b(x-h) \) shifts RIGHT, not left
❌ Don't confuse vertical asymptote location after transformations
✓ Do always find domain by setting argument > 0
✓ Do remember the asymptote moves with horizontal shifts
✓ Do check that x-intercept satisfies the original equation