01-10 - Exponentials and Logarithms
Phase: 1 | Subject: 01-10 Prerequisites: 01-09-polynomials.md (algebraic manipulation), 01-06-systems-of-linear-equations.md (solving equations) Next subject: 02-01-angles-and-lines.md
Learning Objectives
By the end of this subject, you will be able to:
- Understand exponential functions and their graphs
- Define logarithms as the inverse of exponentials
- Apply logarithm laws (product, quotient, power)
- Solve exponential equations using logarithms
- Solve logarithmic equations
Core Content
Exponential Functions
An exponential function has the form y = aˣ where a > 0, a ≠ 1.
Key features: - Always passes through (0, 1) because a⁰ = 1 - Horizontal asymptote: y = 0 (the x-axis) - Always positive (y > 0) - a > 1: increasing (growth) - 0 < a < 1: decreasing (decay)
Example: y = 2ˣ y values: 1/4, 1/2, 1, 2, 4, 8, ... (doubles each step for x = -2, -1, 0, 1, 2, 3)
What is a Logarithm?
⚠️ THIS IS CRITICAL — logarithms are the inverse of exponentials and are essential for calculus (logarithmic differentiation), information theory (entropy, mutual information), and ML (log-likelihood, cross-entropy loss). Understanding logarithms deeply now will save you enormous pain later.
A logarithm answers the question: "What power do we raise the base to, to get the number?"
logₐ(b) = c means aᶜ = b
Example: log₂(8) = 3 because 2³ = 8
Example: log₃(81) = 4 because 3⁴ = 81
Common bases: - log₁₀(x) — common logarithm (often written as log(x)) - ln(x) — natural logarithm (base e ≈ 2.718) - log₂(x) — binary logarithm (used in computer science)
Logarithm Laws
1. Product Rule
logₐ(xy) = logₐ(x) + logₐ(y)
Example: log₂(8 × 4) = log₂(8) + log₂(4) = 3 + 2 = 5
Common pitfall: The product rule applies to MULTIPLICATION inside the log — NOT addition. logₐ(x + y) is NOT equal to logₐ(x) + logₐ(y). There is no simple rule for logₐ(x + y).
2. Quotient Rule
logₐ(x/y) = logₐ(x) - logₐ(y)
Example: log₁₀(100/10) = log₁₀(100) - log₁₀(10) = 2 - 1 = 1
3. Power Rule
logₐ(xᵏ) = k · logₐ(x)
Example: log₅(25²) = 2 · log₅(25) = 2 × 2 = 4
4. Change of Base Formula
logₐ(b) = ln(b) / ln(a) = log₁₀(b) / log₁₀(a)
Example: log₂(10) = ln(10) / ln(2) ≈ 2.3026 / 0.6931 ≈ 3.32
Solving Exponential Equations
Method 1: Same base. If both sides can be written as powers of the same base, equate exponents.
Example: Solve 3^(2x - 1) = 27
- 27 = 3³, so 3^(2x - 1) = 3³
- Equate exponents: 2x - 1 = 3
- 2x = 4, x = 2
Method 2: Take logarithms. When you can't easily get the same base, take the log of both sides.
Example: Solve 5ˣ = 12
- Take natural log: ln(5ˣ) = ln(12)
- x · ln(5) = ln(12)
- x = ln(12) / ln(5) ≈ 2.4849 / 1.6094 ≈ 1.54
Solving Logarithmic Equations
Method: Use log laws to combine logs, then convert to exponential.
Example: Solve log₂(x) + log₂(x - 3) = 3
- Combine: log₂(x(x - 3)) = 3
- Exponential: x(x - 3) = 2³ = 8
- x² - 3x - 8 = 0
- x = (3 ± √(9 + 32))/2 = (3 ± √41)/2
- Only positive solution valid: x ≈ (3 + 6.40)/2 ≈ 4.7
Check domain: x > 0 and x - 3 > 0, so x > 3. x ≈ 4.7 ✓
Key Terms
- 01 10 Exponentials And Logarithms
- Change of Base Formula
- Correct: A)
- Correct: B)
- Example 1: Simplify using log laws
- Example 2: Solve exponential equation
- Example 3: Solve logarithmic equation
- Exponential Functions
- Logarithm Laws
- Method 1: Same base.
- Method 2: Take logarithms.
- Power Rule
Worked Examples
Example 1: Simplify using log laws
Simplify: log₃(27) + log₃(9)
- log₃(27) = 3 (because 3³ = 27)
- log₃(9) = 2 (because 3² = 9)
- Sum: 3 + 2 = 5
Or using product rule: log₃(27 × 9) = log₃(243) = 5 (because 3⁵ = 243)
Example 2: Solve exponential equation
Solve: 4ˣ = 32
- Write both as powers of 2: (2²)ˣ = 2⁵
- 2^(2x) = 2⁵
- Equate exponents: 2x = 5
- x = 2.5
Example 3: Solve logarithmic equation
Solve: ln(x + 1) - ln(x - 2) = ln(4)
- Combine: ln((x + 1)/(x - 2)) = ln(4)
- Since ln is one-to-one: (x + 1)/(x - 2) = 4
- x + 1 = 4(x - 2)
- x + 1 = 4x - 8
- 9 = 3x
- x = 3
Quiz
Q1: What does the concept of Change of Base Formula primarily refer to in this subject?
A) A visual representation of Change of Base Formula B) A computational error related to Change of Base Formula C) A historical anecdote about Change of Base Formula D) The definition and application of Change of Base Formula
Correct: D)
- If you chose A: This is incorrect. Change of Base Formula is defined as: the definition and application of change of base formula. The other options describe different aspects that are not the primary focus.
- If you chose B: This is incorrect. Change of Base Formula is defined as: the definition and application of change of base formula. The other options describe different aspects that are not the primary focus.
- If you chose C: This is incorrect. Change of Base Formula is defined as: the definition and application of change of base formula. The other options describe different aspects that are not the primary focus.
- If you chose D: Change of Base Formula is defined as: the definition and application of change of base formula. The other options describe different aspects that are not the primary focus. Correct!
Q2: What is the primary purpose of Exponential Functions?
A) It replaces all other methods in this domain B) It is used to exponential functions in mathematical analysis C) It is used only in advanced research contexts D) It is primarily a historical notation system
Correct: B)
- If you chose A: This is incorrect. Exponential Functions serves the purpose described in the correct answer. The other options misrepresent its role.
- If you chose B: Exponential Functions serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
- If you chose C: This is incorrect. Exponential Functions serves the purpose described in the correct answer. The other options misrepresent its role.
- If you chose D: This is incorrect. Exponential Functions 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 mentioned only as a historical footnote B) Logarithm Laws is a fundamental concept covered in this subject C) Logarithm Laws is an advanced topic beyond this subject's scope D) Logarithm Laws is not related to this subject
Correct: B)
- 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: 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 C: 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 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) "What power do we raise the base to, to get the nu B) The inverse of the correct answer C) A different result from a common mistake D) An unrelated numerical value
Correct: A)
- If you chose A: The worked examples show that the result is "What power do we raise the base to, to get the nu. The other options represent common errors. Correct!
- If you chose B: This is incorrect. The worked examples show that the result is "What power do we raise the base to, to get the nu. The other options represent common errors.
- If you chose C: This is incorrect. The worked examples show that the result is "What power do we raise the base to, to get the nu. The other options represent common errors.
- If you chose D: This is incorrect. The worked examples show that the result is "What power do we raise the base to, to get the nu. The other options represent common errors.
Q5: How are Logarithm Laws and Power Rule related?
A) Logarithm Laws and Power Rule are completely unrelated topics B) Logarithm Laws is a special case of Power Rule C) Logarithm Laws is the inverse of Power Rule D) Logarithm Laws and Power Rule are closely related concepts
Correct: D)
- If you chose A: This is incorrect. Both Logarithm Laws and Power Rule are covered in this subject as interconnected topics.
- If you chose B: This is incorrect. Both Logarithm Laws and Power Rule are covered in this subject as interconnected topics.
- If you chose C: This is incorrect. Both Logarithm Laws and Power Rule are covered in this subject as interconnected topics.
- If you chose D: Both Logarithm Laws and Power Rule are covered in this subject as interconnected topics. Correct!
Q6: What is a common pitfall when working with What Is A Logarithm??
A) What Is A Logarithm? has no common misconceptions B) A common mistake is confusing What Is A Logarithm? with a similar concept C) What Is A Logarithm? is always computed the same way in all contexts D) The main error with What Is A Logarithm? is using it when it is not needed
Correct: B)
- If you chose A: This is incorrect. Students often confuse What Is A Logarithm? with similar-sounding or related concepts. Pay attention to the precise definitions.
- If you chose B: Students often confuse What Is A Logarithm? with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
- If you chose C: This is incorrect. Students often confuse What Is A Logarithm? with similar-sounding or related concepts. Pay attention to the precise definitions.
- If you chose D: This is incorrect. Students often confuse What Is A Logarithm? with similar-sounding or related concepts. Pay attention to the precise definitions.
Q7: When should you apply Product Rule?
A) Apply Product Rule to solve problems in this subject's domain B) Use Product Rule only in pure mathematics contexts C) Product Rule is not practically useful D) Avoid Product Rule unless explicitly instructed
Correct: A)
- If you chose A: Product Rule is a practical tool used throughout this subject to solve relevant problems. Correct!
- If you chose B: This is incorrect. Product Rule is a practical tool used throughout this subject to solve relevant problems.
- If you chose C: This is incorrect. Product Rule is a practical tool used throughout this subject to solve relevant problems.
- If you chose D: This is incorrect. Product Rule is a practical tool used throughout this subject to solve relevant problems.
Practice Problems
- Evaluate: log₂(32)
Click for answer
5 (because 2⁵ = 32)
- Simplify: log₅(25) + log₅(5)
Click for answer
2 + 1 = 3
- Simplify: log₃(81) - log₃(9)
Click for answer
4 - 2 = 2
- Solve: 3ˣ = 81
Click for answer
x = 4
- Solve: log₄(x) = 2
Click for answer
x = 4² = 16
- Solve: 2^(x + 1) = 10
Click for answer
x + 1 = log₂(10) ≈ 3.322, so x ≈ 2.322
Summary
Key takeaways:
- Exponential: y = aˣ, always positive, asymptote y = 0
- Logarithm: inverse of exponential; logₐ(b) = c means aᶜ = b
- Product rule: log(xy) = log(x) + log(y)
- Quotient rule: log(x/y) = log(x) - log(y)
- Power rule: log(xᵏ) = k·log(x)
- Change of base: logₐ(b) = ln(b)/ln(a)
- Solve exponentials: take logs of both sides or get same base
- Solve logs: combine, convert to exponential
Pitfalls
- Applying the log product rule to addition inside the log. log(x + y) is NOT equal to log(x) + log(y). The product rule only applies to multiplication: log(xy) = log(x) + log(y). There is no simple rule for log(x + y).
- Confusing the meaning of log notation. log₂(8) = 3 means "2 to what power gives 8?" — the answer is the exponent. A common error is reading log₂(8) as 2 × 8 or 2⁸. The logarithm IS the exponent.
- Forgetting to check domain restrictions when solving logarithmic equations. The argument of any logarithm must be positive. For log₂(x) + log₂(x - 3) = 3, both x > 0 AND x - 3 > 0 (so x > 3). Solutions that violate the domain must be discarded even if they satisfy the algebra.
- Dividing instead of subtracting when using the quotient rule. log(x/y) = log(x) - log(y), not log(x)/log(y). Logarithm rules transform products to sums, quotients to differences, and powers to coefficients — but they don't turn division of logs into division of numbers.
- Forgetting the change of base formula when bases don't match. log₂(10) cannot be evaluated directly on most calculators. Use log₂(10) = ln(10)/ln(2) or log₁₀(10)/log₁₀(2). Without the change of base formula, you're stuck.
Next Steps
Next up: 02-01-angles-and-lines.md