Exponential Logarithmic Graphs

Grade 11 · algebra · 100 practice problems · read aloud

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Exponential & Logarithmic Graphs

📈 What Are They & Why Are They Useful?

Exponential and logarithmic functions are inverses of each other. Exponential functions model rapid growth (like populations or investments) or decay (like radioactive material). Logarithmic functions help us solve for the exponent in those models, answering questions like "How long until my investment doubles?"

🛠️ Step-by-Step Guide to Graphing

  1. Identify the Base Function: Is it exponential (y = a⋅bˣ) or logarithmic (y = a + b⋅log(x))?
  2. Find Key Points:
    • Exponential: The y-intercept is always (0, a). Calculate 2-3 more points (e.g., x=1, x=2).
    • Logarithmic: The x-intercept is found by setting y=0. The vertical asymptote is usually x=0.
  3. Plot & Draw: Plot your points, sketch the curve, and label any asymptotes.

📚 Worked Examples

Example 1: Graph y = 2ˣ

Step 1: This is exponential. The y-intercept is (0, 1).
Step 2: Find points: If x=1, y=2. If x=2, y=4. If x=-1, y=1/2.
Step 3: Plot these points. The graph curves upward, with a horizontal asymptote at y=0.

Example 2: Graph y = log₂(x)

Step 1: This is logarithmic. The base 2 log asks "2 to what power equals x?"
Step 2: Find points: If x=1, y=0. If x=2, y=1. If x=4, y=2. If x=1/2, y=-1.
Step 3: Plot these points. The graph curves slowly upward, with a vertical asymptote at x=0.

⚠️ Common Mistakes to Avoid

  • Confusing Growth & Decay: For y = a⋅bˣ, if b > 1, it's growth. If 0 < b < 1, it's decay.
  • Misplacing Asymptotes: Exponential graphs have horizontal asymptotes. Logarithmic graphs have vertical asymptotes.
  • Domain Errors: You cannot take the log of a negative number or zero. The domain of logₐ(x) is x > 0.

💡 Tips & Tricks

  • They Are Mirrors! The graphs of y = bˣ and y = logₐ(x) are reflections over the line y = x.
  • Point Swapping: If (a, b) is on an exponential graph, then (b, a) is on its inverse logarithmic graph.
  • Use Technology: Use a graphing calculator or app like Desmos to visualize the curves and check your work.

🎯 How to Practice

Start by graphing basic functions like y=2ˣ and y=log₂(x) by hand. Then, practice with transformations: y = 2ˣ⁺¹ - 3 or y = log₃(x - 2). Find and label intercepts and asymptotes every time. Finally, tackle word problems that connect these graphs to real-world scenarios like compound interest or sound intensity (decibels).

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 log₃(27) + 2² = ?

Hint: Remember that logarithms ask 'what power gives this result?' and exponents indicate repeated multiplication.

Show the answer

Answer: 7

  1. Evaluate log₃(27). This asks '3 to what power equals 27?' Since 3³ = 27, log₃(27) = 3.
  2. Evaluate 2². This means 2 × 2 = 4.
  3. Add the results: 3 + 4 = 7.

The answer is 7.

2 log₃(81) - 2² = ?

Hint: Consider what power you need to raise the base to get the argument, then evaluate the exponent term separately before combining the results.

Show the answer

Answer: 0

  1. Evaluate log₃(81). Since 3⁴ = 81, log₃(81) = 4.
  2. Evaluate 2² = 4.
  3. Subtract the results: 4 - 4 = 0.

The answer is 0.

3 log₃(81) - 2⁴ = ?

Hint: First evaluate the logarithmic expression by finding what power of the base gives the argument, then calculate the exponential term before performing the subtraction.

Show the answer

Answer: -12

  1. Evaluate log₃(81). Since 3⁴ = 81, log₃(81) = 4.
  2. Evaluate 2⁴. 2 × 2 × 2 × 2 = 16.
  3. Subtract the results: 4 - 16 = -12.

The answer is -12.

4 log₂(32) + ln(e³) = ?

Hint: Remember that logarithms and exponents are inverse operations. For example, log₃(27) = 3 because 3³ = 27, and ln(e⁵) = 5 because e⁵ is the exponential form.

Show the answer

Answer: 8

  1. Evaluate log₂(32) Since 2⁵ = 32, log₂(32) = 5
  2. Evaluate ln(e³) Since ln is the natural logarithm (base e), and e³ is the exponential form, ln(e³) = 3
  3. Add the results 5 + 3 = 8

The answer is 8.

5 log₂(64) + ln(e⁴) = ?

Hint: Remember that logarithms are the inverse of exponential functions. For example, log₃(27) asks '3 to what power equals 27?'

Show the answer

Answer: 10

  1. Evaluate log₂(64). Since 2^6 = 64, log₂(64) = 6.
  2. Evaluate ln(e⁴). Since ln is log base e, and e^4 = e⁴, ln(e⁴) = 4.
  3. Add the results: 6 + 4 = 10.

The answer is 10.

6 log₂(64) + ln(e⁵) = ?

Hint: Remember that logarithms are the inverse of exponential functions. For example, log₃(27) asks '3 to what power equals 27?'

Show the answer

Answer: 11

  1. Evaluate log₂(64). Since 2^6 = 64, log₂(64) = 6.
  2. Evaluate ln(e⁵). Since ln is log base e, and e^5 = e^5, ln(e⁵) = 5.
  3. Add the results: 6 + 5 = 11.

The answer is 11.

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