📐 Concept diagram

03-05 - Exponential and Logarithmic Functions

Phase: 3 | Subject: 03-05 Prerequisites: 03-04-rational-functions.md (asymptotes, domain), 01-10-exponentials-and-logarithms.md (basic exponent/log rules) Next subject: 03-06-complex-numbers.md

Learning Objectives

By the end of this subject, you will be able to:

1. Graph exponential functions aˣ and identify key features
2. Graph logarithmic functions logₐ(x) and identify key features
3. Apply transformations to exponential and logarithmic graphs
4. Solve exponential and logarithmic equations
5. Model exponential growth and decay, including logistic growth

Core Content

Exponential Functions

Form: f(x) = aˣ where a > 0, a ≠ 1

Key features: - Always passes through (0, 1) because a⁰ = 1 - Horizontal asymptote: y = 0 (the x-axis) - Domain: all real numbers - Range: y > 0 - a > 1: growth function (increasing) - 0 < a < 1: decay function (decreasing)

Example: f(x) = 2ˣ f(0) = 1, f(1) = 2, f(2) = 4, f(-1) = 1/2, f(-2) = 1/4

Transformation: y = a·bˣ + k - a: vertical stretch/reflection - b > 1: growth; 0 < b < 1: decay - k: vertical shift

Logarithmic Functions

Form: f(x) = logₐ(x) where a > 0, a ≠ 1

Logarithm is the INVERSE of exponential: y = logₐ(x) means aʸ = x

Key features: - Always passes through (1, 0) because logₐ(1) = 0 - Vertical asymptote: x = 0 (the y-axis) - Domain: x > 0 - Range: all real numbers - a > 1: increasing; 0 < a < 1: decreasing

Example: f(x) = log₂(x) f(1) = 0, f(2) = 1, f(4) = 2, f(1/2) = -1, f(1/4) = -2

Logarithm Laws

$logₐ(xy) = logₐ(x) + logₐ(y)           (product)
logₐ(x/y) = logₐ(x) - logₐ(y)           (quotient)
logₐ(xⁿ) = n·logₐ(x)                    (power)
logₐ(a) = 1                             (identity)
logₐ(1) = 0                             (identity)
logₐ(x) = ln(x) / ln(a)                 (change of base)
a^(logₐ(x)) = x                          (inverse — exponential undoes log)
$

Solving Exponential Equations

Strategy: Get the same base on both sides, then equate exponents.

Example: Solve 3ˣ = 81

3ˣ = 3⁴ x = 4

Example: Solve 2^(2x - 1) = 8

2^(2x - 1) = 2³ 2x - 1 = 3 2x = 4 x = 2

Solving Logarithmic Equations

Strategy: Use log laws to combine or isolate the log, then convert to exponential.

Example: Solve log₂(x) + log₂(x - 3) = 3

1. Combine: log₂(x(x - 3)) = 3
2. Exponential: x(x - 3) = 2³ = 8
3. x² - 3x - 8 = 0
4. x = (3 ± √(9 + 32))/2 = (3 ± √41)/2
5. Only positive solution valid (domain x > 0 and x > 3): x = (3 + √41)/2 ≈ 4.7

Exponential Growth and Decay Models

General form: P(t) = P₀·aᵗ

Continuous model: P(t) = P₀·e^(kt) - k > 0: growth - k < 0: decay

Half-life: Time for quantity to halve Doubling time: Time for quantity to double

Example: Radioactive decay with half-life 5 years. P(t) = P₀·(1/2)^(t/5)

Logistic Growth:

$P(t) = L / (1 + Ae^(-kt))
$

where L = carrying capacity (maximum population) Initially exponential, then slows as it approaches L.

Key Terms

• 03 05 Exponential And Logarithmic Functions
• Correct: A)
• Correct: B)
• Example 1: Graph transformations
• Example 2: Solve exponential
• Exponential Functions
• Exponential Growth and Decay Models
• Logarithm Laws
• Logarithmic Functions
• Solving Exponential Equations
• Solving Logarithmic Equations

Worked Examples

Example 1: Graph transformations

Sketch y = 2·3^(x - 1) + 1

1. Start with y = 3ˣ (passes through (0, 1), asymptote y = 0)
2. Shift right 1: passes through (1, 1)
3. Stretch vertically by 2: passes through (1, 2)
4. Shift up 1: passes through (1, 3), asymptote y = 1

Example 2: Solve exponential

Solve 5^(x + 1) = 3^(2x - 1)

1. Take log of both sides: (x + 1)ln(5) = (2x - 1)ln(3)
2. x·ln(5) + ln(5) = 2x·ln(3) - ln(3)
3. x·ln(5) - 2x·ln(3) = -ln(3) - ln(5)
4. x(ln(5) - 2ln(3)) = -(ln(3) + ln(5))
5. x = -(ln(3) + ln(5)) / (ln(5) - 2ln(3))
6. x ≈ -(1.099 + 1.609) / (1.609 - 2.197) = -2.708 / (-0.588) ≈ 4.61

Quiz

Q1: What does the concept of Exponential Functions primarily refer to in this subject?

A) A historical anecdote about Exponential Functions B) A visual representation of Exponential Functions C) The definition and application of Exponential Functions D) A computational error related to Exponential Functions

Correct: C)

• If you chose A: This is incorrect. Exponential Functions is defined as: the definition and application of exponential functions. The other options describe different aspects that are not the primary focus.
• If you chose B: This is incorrect. Exponential Functions is defined as: the definition and application of exponential functions. The other options describe different aspects that are not the primary focus.
• If you chose C: Exponential Functions is defined as: the definition and application of exponential functions. The other options describe different aspects that are not the primary focus. Correct!
• If you chose D: This is incorrect. Exponential Functions is defined as: the definition and application of exponential functions. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of Exponential Growth and Decay Models?

A) It is used to exponential growth and decay models in mathematical analysis B) It is used only in advanced research contexts C) It is primarily a historical notation system D) It replaces all other methods in this domain

Correct: A)

• If you chose A: Exponential Growth and Decay Models serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose B: This is incorrect. Exponential Growth and Decay Models serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose C: This is incorrect. Exponential Growth and Decay Models serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose D: This is incorrect. Exponential Growth and Decay Models serves the purpose described in the correct answer. The other options misrepresent its role.

Q3: Which statement about Logarithm Laws is TRUE?

A) Logarithm Laws is an advanced topic beyond this subject's scope B) Logarithm Laws is mentioned only as a historical footnote C) Logarithm Laws is a fundamental concept covered in this subject D) Logarithm Laws is not related to this subject

Correct: C)

• If you chose A: This is incorrect. Logarithm Laws is a fundamental concept covered in this subject. This subject covers Logarithm Laws as part of its core content.
• If you chose B: This is incorrect. Logarithm Laws is a fundamental concept covered in this subject. This subject covers Logarithm Laws as part of its core content.
• If you chose C: Logarithm Laws is a fundamental concept covered in this subject. This subject covers Logarithm Laws as part of its core content. Correct!
• If you chose D: This is incorrect. Logarithm Laws is a fundamental concept covered in this subject. This subject covers Logarithm Laws as part of its core content.

Q4: Based on the worked examples in this subject, what is the correct result?

A) 0, increasing, domain ℝ, range (0, ∞). B) An unrelated numerical value C) A different result from a common mistake D) The inverse of the correct answer

Correct: A)

• If you chose A: The worked examples show that the result is 0, increasing, domain ℝ, range (0, ∞).. The other options represent common errors. Correct!
• If you chose B: This is incorrect. The worked examples show that the result is 0, increasing, domain ℝ, range (0, ∞).. The other options represent common errors.
• If you chose C: This is incorrect. The worked examples show that the result is 0, increasing, domain ℝ, range (0, ∞).. The other options represent common errors.
• If you chose D: This is incorrect. The worked examples show that the result is 0, increasing, domain ℝ, range (0, ∞).. The other options represent common errors.

Q5: How are Logarithm Laws and Logarithmic Functions related?

A) Logarithm Laws and Logarithmic Functions are completely unrelated topics B) Logarithm Laws is the inverse of Logarithmic Functions C) Logarithm Laws is a special case of Logarithmic Functions D) Logarithm Laws and Logarithmic Functions are closely related concepts

Correct: D)

• If you chose A: This is incorrect. Both Logarithm Laws and Logarithmic Functions are covered in this subject as interconnected topics.
• If you chose B: This is incorrect. Both Logarithm Laws and Logarithmic Functions are covered in this subject as interconnected topics.
• If you chose C: This is incorrect. Both Logarithm Laws and Logarithmic Functions are covered in this subject as interconnected topics.
• If you chose D: Both Logarithm Laws and Logarithmic Functions are covered in this subject as interconnected topics. Correct!

Q6: What is a common pitfall when working with Solving Exponential Equations?

A) A common mistake is confusing Solving Exponential Equations with a similar concept B) Solving Exponential Equations has no common misconceptions C) The main error with Solving Exponential Equations is using it when it is not needed D) Solving Exponential Equations is always computed the same way in all contexts

Correct: A)

• If you chose A: Students often confuse Solving Exponential Equations with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose B: This is incorrect. Students often confuse Solving Exponential Equations with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose C: This is incorrect. Students often confuse Solving Exponential Equations with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose D: This is incorrect. Students often confuse Solving Exponential Equations with similar-sounding or related concepts. Pay attention to the precise definitions.

Q7: When should you apply Solving Logarithmic Equations?

A) Solving Logarithmic Equations is not practically useful B) Use Solving Logarithmic Equations only in pure mathematics contexts C) Avoid Solving Logarithmic Equations unless explicitly instructed D) Apply Solving Logarithmic Equations to solve problems in this subject's domain

Correct: D)

• If you chose A: This is incorrect. Solving Logarithmic Equations is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: This is incorrect. Solving Logarithmic Equations is a practical tool used throughout this subject to solve relevant problems.
• If you chose C: This is incorrect. Solving Logarithmic Equations is a practical tool used throughout this subject to solve relevant problems.
• If you chose D: Solving Logarithmic Equations is a practical tool used throughout this subject to solve relevant problems. Correct!

Practice Problems

1. Graph features of y = 4ˣ Answer: Passes through (0, 1), asymptote y = 0, increasing, domain ℝ, range (0, ∞).

2. Solve: log₃(x) = 4 Answer: x = 3⁴ = 81

3. Solve: e^(2x) = 7 Answer: 2x = ln(7), x = ln(7)/2 ≈ 0.973

4. Simplify: ln(8) - ln(2) Answer: ln(8/2) = ln(4)

5. If P(t) = 100·e^(0.05t), what is P after 10 years? Answer: P(10) = 100·e^(0.5) ≈ 100 × 1.649 = 164.9

Summary

Key takeaways:

• Exponential aˣ: domain ℝ, range (0, ∞), asymptote y = 0
• Logarithm logₐ(x): domain (0, ∞), range ℝ, asymptote x = 0
• Log laws: product, quotient, power, change of base
• Solve exponentials: get same base or use logs
• Solve logs: combine, convert to exponential
• Growth/decay models: P(t) = P₀·e^(kt)

Pitfalls

• Assuming log(x + y) = log x + log y: The product rule gives log(xy) = log x + log y, but log(x + y) has no simplification. Students frequently extend the log product rule to sums, which is incorrect. Only multiplication inside the log splits into a sum of logs.
• Forgetting to check the domain when solving log equations: After solving an equation like log₂(x) + log₂(x - 3) = 3, you must verify that your solutions satisfy x > 0 and x > 3 (since the argument of a logarithm must be positive). Extraneous solutions can arise from algebraic manipulation — always check against the original domain.
• Misapplying the change of base formula: logₐ(x) = ln(x) / ln(a), not ln(x / a). Students sometimes write logₐ(x) = ln(x) / a, forgetting the ln around the base. The denominator is the log of the BASE, not the base itself.
• Confusing exponential asymptotes: An exponential function like y = 2ˣ has horizontal asymptote y = 0, while its inverse, the logarithmic function y = log₂(x), has vertical asymptote x = 0. Students often swap these or forget that the asymptote shifts under transformations (e.g., y = 2ˣ + 3 shifts the asymptote to y = 3).
• Taking the log of both sides incorrectly: When solving 5^(x+1) = 3^(2x-1), you take ln of the whole LHS and RHS, but the power rule moves the exponent DOWN: (x+1)ln 5 = (2x-1)ln 3. A common mistake is writing 5^(x+1)ln 5 instead of (x+1)ln 5.

Next Steps

Next up: 03-06-complex-numbers.md