Session 01-05 - Logarithms & Substitution
Section 01: Mathematical Foundations & Algebra
Entry Quiz
Quick Review from Last Session
Complete individually, then we discuss
Factor completely: \(x^3 - 27\)
Simplify: \(\sqrt{48} + \sqrt{12} - \sqrt{75}\)
Rationalize: \(\frac{3}{\sqrt{5} - 2}\)
Factor using AC method: \(2x^2 + 7x + 3\)
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Let’s review together!
Homework Discussion - 15 Minutes
Homework Showcase
15 minutes for presentations and discussion
- Present and discuss your solutions from Tasks 01-04
- Share any challenging problems or interesting approaches
- Use this time to clarify concepts before we move forward
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Today we build on factorization and radicals with new powerful tools!
Learning Objectives
What You’ll Master Today
- Use substitution to factor expressions with hidden quadratic structure
- Explain logarithms as the inverse of exponentiation
- Apply the logarithm laws to simplify and evaluate expressions
- Solve exponential equations for time and growth questions in business
- Expand binomial powers using Pascal’s triangle
Substitution for Factorization
What is Substitution for Factorization?
Making complex expressions simpler by introducing a new variable
Sometimes factorization becomes easier when we substitute part of an expression with a simpler variable.
- Strategy: Replace a repeated expression with a single variable
- Factor the simpler expression
- Substitute back to get the final answer
- Why it works: Reduces cognitive load and reveals hidden patterns
. . .
Look for expressions that appear multiple times or have a clear “inner” structure!
When to Use Substitution
Recognize these common patterns
- Quadratic in form: \((x^2)^2 + 5(x^2) + 6\)
- Repeated expressions: \((2x + 1)^2 - 3(2x + 1) - 10\)
- Complex nested terms: \(\sqrt{x + 1} - 2\sqrt{x + 1} + 1\)
- Trigonometric expressions: \(\sin^2(x) + 3\sin(x) + 2\)
. . .
The key is identifying what to substitute - look for the “building block” that repeats!
Substitution Examples
Factor \(x^4 - 13x^2 + 36\)
- Step 1: Let \(u = x^2\), so \(x^4 = u^2\)
- Step 2: Substitute: \(u^2 - 13u + 36\)
- Step 3: Factor: \((u - 4)(u - 9)\)
- Step 4: Substitute back: \((x^2 - 4)(x^2 - 9)\)
- Step 5: Factor completely: \((x - 2)(x + 2)(x - 3)(x + 3)\)
Factor \(x + 6\sqrt{x} + 8\)
- Step 1: Notice this involves \(x\) and \(\sqrt{x}\), where \(x = (\sqrt{x})^2\)
- Step 2: Let \(u = \sqrt{x}\), so \(x = u^2\)
- Step 3: Substitute: \(u^2 + 6u + 8\)
- Step 4: Factor: \((u + 2)(u + 4)\)
- Step 5: Substitute back: \((\sqrt{x} + 2)(\sqrt{x} + 4)\)
Factor \(3x^6 - 11x^3 - 20\)
- Step 1: Let \(u = x^3\), so \(x^6 = u^2\) and we have \(3u^2 - 11u - 20\)
- Step 2: Use AC method: \(ac = 3(-20) = -60\)
- Step 3: Find factors of -60 that sum to -11: (4, -15)
- Step 4: Rewrite: \(3u^2 + 4u - 15u - 20\)
- Step 5: Group: \(u(3u + 4) - 5(3u + 4) = (u - 5)(3u + 4)\)
- Step 6: Substitute back: \((x^3 - 5)(3x^3 + 4)\)
Common Substitutions
Simplification tricks
When you see:
- \(3^{2x}\) → Let \(u = 3^x\), then \(3^{2x} = u^2\)
- \(\sqrt{x}\) appearing multiple times → Let \(u = \sqrt{x}\)
- Symmetric expressions → Look for factoring patterns
- Repeating decimals → Use algebraic method to find fraction
Practice with Substitution
Work individually for 8 minutes
Factor: \(x^6 + 8x^3 + 16\)
Factor: \((\sqrt{x} - 2)^2 - 5(\sqrt{x} - 2) + 6\)
Factor: \(16x^4 - 81\)
Factor: \((x^2 + 3x)^2 - 8(x^2 + 3x) + 15\)
. . .
Always check if you can factor further after substituting back!
More Advanced Substitution
Sometimes you need to think a little bit more
Example: Factor \(x^{2/3} - 5x^{1/3} + 6\)
- Observation: This involves fractional exponents
- Key insight: Let \(u = x^{1/3}\), so \(x^{2/3} = (x^{1/3})^2 = u^2\)
- Step 1: Substitute: \(u^2 - 5u + 6\)
- Step 2: Factor: \((u - 2)(u - 3)\)
- Step 3: Substitute back: \((x^{1/3} - 2)(x^{1/3} - 3)\)
- Step 4: Can also write as: \((\sqrt[3]{x} - 2)(\sqrt[3]{x} - 3)\)
Break - 10 Minutes
Logarithms - The Basics
What is a Logarithm?
The logarithm is the inverse of exponentiation
\[\text{If } a^x = b \text{, then } \log_a(b) = x\]
Think of it as: “What power do I raise \(a\) to get \(b\)?”
- \(2^3 = 8\) means \(\log_2(8) = 3\)
- \(5^x = 125\) means \(x = \log_5(125) = 3\)
. . .
Standard notation:
- \(\log\) without a base means \(\log_{10}\) (common logarithm)
- \(\ln\) means \(\log_e\) where \(e \approx 2.718\) (natural logarithm)
The Logarithm is a Mirror Image
\(\log_2(x)\) undoes \(2^x\) - graphically, a reflection across \(y = x\)
Key Logarithm Properties
These follow directly from exponent laws!
| Property | Formula | Why it works |
|---|---|---|
| \(\log_a(1) = 0\) | Because \(a^0 = 1\) | Any base to the 0 is 1 |
| \(\log_a(a) = 1\) | Because \(a^1 = a\) | Base to the 1st is itself |
| \(\log_a(a^x) = x\) | Direct from definition | Inverse operations |
| \(a^{\log_a(x)} = x\) | Direct from definition | Inverse operations |
Important: Logarithms are transcendental functions - they cannot be expressed using only algebraic operations (unlike polynomials, radicals, and rational functions).
Laws of Logarithms
These transform complex operations into simple ones
| Rule | Formula | Example |
|---|---|---|
| Product | \(\log_a(xy) = \log_a(x) + \log_a(y)\) | \(\log(20) = \log(4) + \log(5)\) |
| Quotient | \(\log_a(\frac{x}{y}) = \log_a(x) - \log_a(y)\) | \(\log(\frac{100}{4}) = \log(100) - \log(4)\) |
| Power | \(\log_a(x^n) = n\log_a(x)\) | \(\log(8^3) = 3\log(8)\) |
. . .
WarningCommon Mistake
\(\log(x + y) \neq \log(x) + \log(y)\)
There’s NO simple rule for \(\log(x + y)\)!
Spot the Error
Can you find the errors? Work with your neighbor
Time allocation: 5 minutes to find errors, 5 minutes to discuss
Student work:
“\(\log_2(8) + \log_2(4) = \log_2(12)\)”
“\(\frac{\log(100)}{\log(10)} = \log\left(\frac{100}{10}\right) = \log(10) = 1\)”
“\(\log_3(9^x) = (\log_3(9))^x = 2^x\)”
Working with Logarithms
Evaluating Logarithms
Find \(\log_3(81)\)
- Ask: “3 to what power equals 81?”
- \(3^1 = 3\), \(3^2 = 9\), \(3^3 = 27\), \(3^4 = 81\)
- Therefore: \(\log_3(81) = 4\)
Simplify \(\log_2(32) + \log_2(8) - \log_2(4)\)
- Method 1: Evaluate each
- \(\log_2(32) = 5\), \(\log_2(8) = 3\), \(\log_2(4) = 2\)
- Result: \(5 + 3 - 2 = 6\)
- Method 2: Use laws
- \(= \log_2(\frac{32 \times 8}{4}) = \log_2(64) = 6\)
Change of Base Formula
Convert between different bases
\[\log_a(x) = \frac{\log_b(x)}{\log_b(a)} = \frac{\ln(x)}{\ln(a)} = \frac{\log(x)}{\log(a)}\]
. . .
Example: Find \(\log_5(30)\)
- \(\log_5(30) = \frac{\ln(30)}{\ln(5)} = \frac{3.401}{1.609} \approx 2.113\)
- Check: \(5^{2.113} \approx 30\) ✓
Calculator: Logarithms
Your calculator handles all three logarithm types:
| Function | Access |
|---|---|
| \(\log_{10}(x)\) | SHIFT + (-) button |
| \(\ln(x)\) | ln button |
| \(\log_a(b)\) | log (log button) |
. . .
Example: Calculate \(\log_2(32)\)
. . .
Enter: log 2 → 32 =
. . .
Result: 5 (because \(2^5 = 32\))
Individual Exercise 01
Work individually for 5 minutes
Evaluate: \(\log_4(64)\)
Simplify: \(\log_3(9) + \log_3(27)\)
Solve: \(\log_5(x + 4) = 2\)
Express as a single logarithm: \(2\log(x) - \log(y) + \log(3)\)
Logarithms in the Real World
Why Logarithms Matter
From protecting your hearing to predicting disasters
- The Challenge: Natural phenomena span enormous ranges
- Human perception: We sense changes proportionally, not linearly
- The Solution: Logarithmic scales compress huge ranges into manageable numbers
- Real Impact: These scales help save lives and advance science
. . .
Historical Note: Logarithms were invented in 1614 by John Napier to simplify astronomical calculations. Today, they’re essential for measuring everything from sound to earthquakes!
How Slowly Does a Logarithm Grow?
While \(x\) runs from 1 to 1,000, \(\log_{10}(x)\) only climbs from 0 to 3
Scientific Applications
Sound intensity spans 24 orders of magnitude - hence the decibel scale: \(L = 10\log(\frac{I}{I_0})\) dB
- Whisper: 30 dB (1,000× threshold)
- Normal conversation: 60 dB (1,000,000× threshold)
- Rock concert: 110 dB (100,000,000,000× threshold)
Each 10 dB step = 10× intensity - a rock concert is 1,000× a conversation!
The problem with linear scales: Earthquake energy ranges from equivalents of small explosions to thousands of atomic bombs!
Richter scale: \(M = \log_{10}(\frac{A}{A_0})\)
- Magnitude 3: Barely felt (like a large truck passing)
- Magnitude 5: Light damage (100× stronger than Mag 3)
- Magnitude 7: Major earthquake (10,000× stronger than Mag 3)
- Magnitude 9: Great earthquake (1,000,000× stronger than Mag 3)
Financial Applications
Why Natural Logarithm for Finance?
The connection to continuous growth
- Any logarithm works: \(t = \frac{\log_{10}(2)}{\log_{10}(1 + r)} = \frac{\ln(2)}{\ln(1 + r)}\)
- But ln is natural because it connects to continuous compounding
- Continuous compounding formula: \(A = Pe^{rt}\) (where \(e \approx 2.718\))
- Why e appears: It’s the limit as compounding frequency → infinity
Compound Interest Time Calculations
How long to double your money?
- Formula: \(2P = P(1 + r)^t\)
- Simplify: \(2 = (1 + r)^t\)
- Take logarithms: \(\ln(2) = t \cdot \ln(1 + r)\)
- Solve: \(t = \frac{\ln(2)}{\ln(1 + r)}\)
. . .
Rule of 72: At r% interest, doubling time ≈ \(\frac{72}{r}\) years
Practice: Reaching a Savings Goal
Work individually for 5 minutes
You invest 5,000 EUR at 4% interest per year, compounded annually.
Write down the equation for the value of the investment after \(t\) years.
How long until the investment reaches 8,000 EUR?
Use the Rule of 72 to estimate the doubling time, then compare it with the exact value from logarithms.
Coffee Break - 15 Minutes
Advanced Algebraic Techniques
Solving Exponential Equations
Logarithms are the key tool
Solve \(3^{2x-1} = 81\)
- Recognize: \(81 = 3^4\)
- So: \(3^{2x-1} = 3^4\)
- Therefore: \(2x - 1 = 4\)
- Solve: \(x = 2.5\)
Solve \(5^x = 30\)
- Take logarithms: \(\log(5^x) = \log(30)\)
- Use power rule: \(x \cdot \log(5) = \log(30)\)
- Solve: \(x = \frac{\log(30)}{\log(5)} \approx 2.113\)
Expanding Binomial Powers
Pascal’s Triangle
A pattern of binomial coefficients
Row 0: 1
Row 1: 1 1
Row 2: 1 2 1
Row 3: 1 3 3 1
Row 4: 1 4 6 4 1
Row 5: 1 5 10 10 5 1
Row 6: 1 6 15 20 15 6 1- Each number = sum of two above
- Row n gives coefficients for \((a + b)^n\)
- Symmetric pattern
Pair Exercise
Work in pairs for 5 minutes
- Expand completely: \((x - 3)^3\)
Individual Exercise
Work individually for 5 minutes
If \(\log_2(x) + \log_4(x) = 3\), find x.
Expand: \((3x - 2y)^3\)
Simplify: \(\log_3(27) - \log_3(3)\)
Solve: \(2^{x+1} = 32\)
Wrap-Up & Key Takeaways
Key Takeaways
- Substitution is a powerful technique for simplifying expressions
- Logarithms are inverse exponentials
- The logarithm laws simplify complex calculations
- Pascal’s triangle gives binomial coefficients
- These tools are essential for calculus, statistics, and finance
Final Assessment
Quick Check
Work individually then we discuss
Evaluate \(\log_5(125)\).
Solve \(2^{3x} = 64\).
Write \(2\log(x) + \log(5)\) as a single logarithm.
True or False: \(\log(a \cdot b) = \log(a) \cdot \log(b)\).
Next Session Preview
Session 01-06 (Synthesis):
- Integration of ALL Section 1 concepts
- Complex problem-solving strategies
- Business case studies
. . .
Start reviewing all Section 1 material - synthesis session next!
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TipHomework Assignment
Complete Tasks 01-05!