Practice Worksheet Graphing Logarithmic Functions Answer Key

Graphing logarithmic functions can seem daunting at first, but with a systematic approach and a solid understanding of the underlying principles, it becomes a manageable and even insightful task. This comprehensive guide will walk you through the process of graphing logarithmic functions, provide practice worksheets to hone your skills, and offer an answer key for self-assessment. We’ll cover essential concepts, transformations, and practical tips to help you master the art of graphing logarithmic functions.

Understanding Logarithmic Functions

A logarithmic function is the inverse of an exponential function. The logarithmic function y = log_b(x) asks the question: "To what power must we raise b to get x?" Here, b is the base of the logarithm, and x is the argument.

Key Properties of Logarithmic Functions

• Domain: The domain of a logarithmic function y = log_b(x) is x > 0. Logarithms are only defined for positive arguments.
• Range: The range of a logarithmic function is all real numbers.
• Vertical Asymptote: Logarithmic functions have a vertical asymptote at x = 0 (for basic forms) because the function approaches negative infinity as x approaches 0 from the right.
• x-intercept: The x-intercept of a logarithmic function y = log_b(x) is always at (1, 0) because log_b(1) = 0 for any base b.
• Base and Behavior: If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.

Basic Logarithmic Function

The simplest logarithmic function is y = log_b(x), where b is a constant base. Commonly used bases include:

• Common Logarithm: Base 10, denoted as y = log(x) or y = log₁₀(x).
• Natural Logarithm: Base e (Euler's number, approximately 2.71828), denoted as y = ln(x) or y = log_e(x).

Steps to Graphing Logarithmic Functions

Graphing logarithmic functions involves identifying key features, plotting points, and sketching the curve. Here’s a step-by-step guide:

1. Identify the Base: Determine the base b of the logarithmic function y = log_b(x). This will influence the general shape of the graph.
2. Find the Vertical Asymptote: For the basic form y = log_b(x), the vertical asymptote is x = 0. If the function is transformed, the asymptote will shift accordingly.
3. Determine the x-intercept: The x-intercept is where the graph crosses the x-axis (y = 0). For the basic form, this is (1, 0).
4. Choose Additional Points: Select a few additional x-values and calculate the corresponding y-values. It’s helpful to choose x-values that are powers of the base b.
5. Plot the Points: Plot the x-intercept and the additional points you calculated.
6. Sketch the Curve: Draw a smooth curve that approaches the vertical asymptote but never touches it, and passes through the plotted points.
7. Consider Transformations: If the function involves transformations (shifts, stretches, reflections), apply these transformations to the basic graph.

Transformations of Logarithmic Functions

Transformations can significantly alter the graph of a logarithmic function. Understanding these transformations is crucial for accurate graphing.

Vertical Shifts

A vertical shift is represented by adding or subtracting a constant k from the logarithmic function:

• y = log_b(x) + k: Shifts the graph k units upwards if k > 0, and k units downwards if k < 0.

Horizontal Shifts

A horizontal shift is represented by adding or subtracting a constant h inside the logarithm:

• y = log_b(x - h): Shifts the graph h units to the right if h > 0, and h units to the left if h < 0. This also shifts the vertical asymptote to x = h.

Vertical Stretches and Compressions

A vertical stretch or compression is represented by multiplying the logarithmic function by a constant a:

• y = a * log_b(x): If |a| > 1, the graph is vertically stretched. If 0 < |a| < 1, the graph is vertically compressed.

Reflections

Reflections involve negating either the logarithmic function or the argument:

• y = -log_b(x): Reflects the graph across the x-axis.
• y = log_b(-x): Reflects the graph across the y-axis.

Example Problems with Step-by-Step Solutions

Let's work through a few examples to illustrate the process of graphing logarithmic functions.

Example 1: Graph y = log₂(x):

1. Base: The base is b = 2.
2. Vertical Asymptote: x = 0.
3. x-intercept: (1, 0).
4. Additional Points:
   • If x = 2, y = log₂(2) = 1 (Point: (2, 1))
   • If x = 4, y = log₂(4) = 2 (Point: (4, 2))
   • If x = 1/2, y = log₂(1/2) = -1 (Point: (1/2, -1))
5. Plot Points and Sketch: Plot the points (1, 0), (2, 1), (4, 2), and (1/2, -1). Sketch a smooth curve that approaches the vertical asymptote x = 0 and passes through these points.

Example 2: Graph y = log₃(x - 1):

1. Base: The base is b = 3.
2. Vertical Asymptote: x = 1 (shifted 1 unit to the right).
3. x-intercept: Set y = 0: 0 = log₃(x - 1) => 3⁰ = x - 1 => 1 = x - 1 => x = 2. The x-intercept is (2, 0).
4. Additional Points:
   • If x = 4, y = log₃(4 - 1) = log₃(3) = 1 (Point: (4, 1))
   • If x = 10, y = log₃(10 - 1) = log₃(9) = 2 (Point: (10, 2))
   • If x = 4/3, y = log₃(4/3 - 1) = log₃(1/3) = -1 (Point: (4/3, -1))
5. Plot Points and Sketch: Plot the points (2, 0), (4, 1), (10, 2), and (4/3, -1). Sketch a smooth curve that approaches the vertical asymptote x = 1 and passes through these points.

Example 3: Graph y = -2 * log(x) + 3:

1. Base: The base is b = 10 (common logarithm).
2. Vertical Asymptote: x = 0.
3. x-intercept: Set y = 0: 0 = -2 * log(x) + 3 => 2 * log(x) = 3 => log(x) = 3/2 => x = 10^3/2 ≈ 31.62. The x-intercept is approximately (31.62, 0).
4. Additional Points:
   • If x = 1, y = -2 * log(1) + 3 = -2 * 0 + 3 = 3 (Point: (1, 3))
   • If x = 10, y = -2 * log(10) + 3 = -2 * 1 + 3 = 1 (Point: (10, 1))
   • If x = 100, y = -2 * log(100) + 3 = -2 * 2 + 3 = -1 (Point: (100, -1))
5. Plot Points and Sketch: Plot the points (1, 3), (10, 1), (100, -1), and (31.62, 0). Sketch a smooth curve that approaches the vertical asymptote x = 0 and passes through these points. Note the reflection across the x-axis and the vertical stretch.

Practice Worksheet: Graphing Logarithmic Functions

Graph each logarithmic function. Identify the base, vertical asymptote, x-intercept, and at least two additional points.

Worksheet 1:

1. y = log₄(x)
2. y = log₅(x)
3. y = log_1/2(x)
4. y = ln(x)
5. y = log(x)

Worksheet 2:

1. y = log₂(x + 3)
2. y = log₃(x - 2)
3. y = log(x) - 4
4. y = ln(x) + 2
5. y = log_1/3(x + 1)

Worksheet 3:

1. y = 2 * log₂(x)
2. y = -log₃(x)
3. y = 0.5 * log(x)
4. y = -3 * ln(x)
5. y = -log₄(x) + 1

Worksheet 4:

1. y = log₂(2x)
2. y = log₃(x/3)
3. y = ln(-x)
4. y = log(10x)
5. y = 2log(x+1)-3

Worksheet 5 (Challenging):

1. y = log₂(4 - x)
2. y = -log₃(x + 2) + 1
3. y = 2ln(x - 1) - 3
4. y = -0.5log(x + 3) + 2
5. y = log|x|

Answer Key

Here’s the answer key for the practice worksheets. Use it to check your work and identify areas where you may need more practice. Remember, understanding the process is more important than just getting the correct answer.

Answer Key for Worksheet 1:

1. y = log₄(x):
   • Base: 4
   • Vertical Asymptote: x = 0
   • x-intercept: (1, 0)
   • Additional Points: (4, 1), (16, 2), (1/4, -1)
2. y = log₅(x):
   • Base: 5
   • Vertical Asymptote: x = 0
   • x-intercept: (1, 0)
   • Additional Points: (5, 1), (25, 2), (1/5, -1)
3. y = log_1/2(x):
   • Base: 1/2
   • Vertical Asymptote: x = 0
   • x-intercept: (1, 0)
   • Additional Points: (1/2, 1), (1/4, 2), (2, -1)
4. y = ln(x):
   • Base: e
   • Vertical Asymptote: x = 0
   • x-intercept: (1, 0)
   • Additional Points: (e, 1) ≈ (2.718, 1), (e², 2) ≈ (7.389, 2), (1/e, -1) ≈ (0.368, -1)
5. y = log(x):
   • Base: 10
   • Vertical Asymptote: x = 0
   • x-intercept: (1, 0)
   • Additional Points: (10, 1), (100, 2), (1/10, -1)

Answer Key for Worksheet 2:

1. y = log₂(x + 3):
   • Base: 2
   • Vertical Asymptote: x = -3
   • x-intercept: (-2, 0)
   • Additional Points: (-1, 1), (1, 2), (-5/2, -1)
2. y = log₃(x - 2):
   • Base: 3
   • Vertical Asymptote: x = 2
   • x-intercept: (3, 0)
   • Additional Points: (5, 1), (11, 2), (7/3, -1)
3. y = log(x) - 4:
   • Base: 10
   • Vertical Asymptote: x = 0
   • x-intercept: (10000, 0)
   • Additional Points: (1, -4), (10, -3), (100, -2)
4. y = ln(x) + 2:
   • Base: e
   • Vertical Asymptote: x = 0
   • x-intercept: (e^-2, 0) ≈ (0.135, 0)
   • Additional Points: (1, 2), (e, 3) ≈ (2.718, 3), (1/e, 1) ≈ (0.368, 1)
5. y = log_1/3(x + 1):
   • Base: 1/3
   • Vertical Asymptote: x = -1
   • x-intercept: (0, 0)
   • Additional Points: (-2/3, 1), (-8/9, 2), (2, -1)

Answer Key for Worksheet 3:

1. y = 2 * log₂(x):
   • Base: 2
   • Vertical Asymptote: x = 0
   • x-intercept: (1, 0)
   • Additional Points: (2, 2), (4, 4), (1/2, -2)
2. y = -log₃(x):
   • Base: 3
   • Vertical Asymptote: x = 0
   • x-intercept: (1, 0)
   • Additional Points: (3, -1), (9, -2), (1/3, 1)
3. y = 0.5 * log(x):
   • Base: 10
   • Vertical Asymptote: x = 0
   • x-intercept: (1, 0)
   • Additional Points: (10, 0.5), (100, 1), (1/10, -0.5)
4. y = -3 * ln(x):
   • Base: e
   • Vertical Asymptote: x = 0
   • x-intercept: (1, 0)
   • Additional Points: (e, -3) ≈ (2.718, -3), (e², -6) ≈ (7.389, -6), (1/e, 3) ≈ (0.368, 3)
5. y = -log₄(x) + 1:
   • Base: 4
   • Vertical Asymptote: x = 0
   • x-intercept: (4, 0)
   • Additional Points: (1, 1), (16, -1), (1/4, 2)

Answer Key for Worksheet 4:

1. y = log₂(2x):
   • Base: 2
   • Vertical Asymptote: x = 0
   • x-intercept: (1/2, 0)
   • Additional Points: (1, 1), (2, 2), (1/4, -1)
2. y = log₃(x/3):
   • Base: 3
   • Vertical Asymptote: x = 0
   • x-intercept: (3, 0)
   • Additional Points: (9, 1), (27, 2), (1, -1)
3. y = ln(-x):
   • Base: e
   • Vertical Asymptote: x = 0
   • x-intercept: (-1, 0)
   • Additional Points: (-e, 1) ≈ (-2.718, 1), (-e², 2) ≈ (-7.389, 2), (-1/e, -1) ≈ (-0.368, -1)
4. y = log(10x):
   • Base: 10
   • Vertical Asymptote: x = 0
   • x-intercept: (1/10, 0)
   • Additional Points: (1, 1), (10, 2), (1/100, -1)
5. y = 2log(x+1)-3:
   • Base: 10
   • Vertical Asymptote: x = -1
   • x-intercept: (30.62, 0)
   • Additional Points: (0, -3), (9, -1), (99, 1)

Answer Key for Worksheet 5:

1. y = log₂(4 - x):
   • Base: 2
   • Vertical Asymptote: x = 4
   • x-intercept: (3, 0)
   • Additional Points: (2, 1), (0, 2), (7/2, -1)
2. y = -log₃(x + 2) + 1:
   • Base: 3
   • Vertical Asymptote: x = -2
   • x-intercept: (-1, 1)
   • Additional Points: (1, 0), (7, -1), (-5/3, 2)
3. y = 2ln(x - 1) - 3:
   • Base: e
   • Vertical Asymptote: x = 1
   • x-intercept: (1 + e^3/2, 0) ≈ (5.482, 0)
   • Additional Points: (1 + e, -1) ≈ (3.718, -1), (1 + e², 1) ≈ (8.389, 1), (1 + e^1/2, -2) ≈ (2.649, -2)
4. y = -0.5log(x + 3) + 2:
   • Base: 10
   • Vertical Asymptote: x = -3
   • x-intercept: (97, 0)
   • Additional Points: (-2, 2), (7, 1.5), (97,0)
5. y = log|x|:
   • Base: 10
   • Vertical Asymptotes: None.
   • x-intercepts: (-1, 0), (1, 0)
   • This function is symmetric about the y-axis.
   • Additional Points: (10, 1), (-10, 1), (100, 2), (-100, 2)

Common Mistakes and How to Avoid Them

• Forgetting the Vertical Asymptote: Always identify the vertical asymptote first. It serves as a boundary for the graph.
• Incorrectly Applying Transformations: Pay close attention to the order of transformations. Horizontal shifts affect the vertical asymptote.
• Confusing Bases: Ensure you are using the correct base for your calculations. Common logarithm (base 10) and natural logarithm (base e) are often confused.
• Not Understanding the Domain: Logarithmic functions are only defined for positive arguments. Remember that x > 0 for y = log_b(x).
• Miscalculating Points: Double-check your calculations when finding additional points. A single mistake can lead to an inaccurate graph.

Tips for Success

• Practice Regularly: The more you practice, the more comfortable you will become with graphing logarithmic functions.
• Use Graphing Tools: Use online graphing calculators or software to visualize the graphs and check your work.
• Understand the Theory: A solid understanding of the underlying principles will make graphing easier and more intuitive.
• Break Down Complex Functions: For complex functions, break them down into simpler transformations and graph them step by step.
• Review Basic Logarithmic Properties: Familiarize yourself with logarithmic properties, such as the change of base formula, to simplify expressions.

Graphing logarithmic functions is a valuable skill that enhances your understanding of mathematics. By following these steps, practicing with the provided worksheets, and learning from your mistakes, you can master this topic and build a strong foundation for more advanced mathematical concepts. Remember to focus on understanding the transformations and key properties, and don't hesitate to use graphing tools to visualize your results. With consistent effort and a systematic approach, you'll be graphing logarithmic functions with confidence in no time!