Session 01-05 - Logarithms & Substitution

Section 01: Mathematical Foundations & Algebra

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz

Quick Review from Last Session

Complete individually, then we discuss

  1. Factor completely: \(x^3 - 27\)

  2. Simplify: \(\sqrt{48} + \sqrt{12} - \sqrt{75}\)

  3. Rationalize: \(\frac{3}{\sqrt{5} - 2}\)

  4. Factor using AC method: \(2x^2 + 7x + 3\)

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

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

  1. Factor: \(x^6 + 8x^3 + 16\)

  2. Factor: \((\sqrt{x} - 2)^2 - 5(\sqrt{x} - 2) + 6\)

  3. Factor: \(16x^4 - 81\)

  4. 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 \(\log_a(x^n) = n\log_a(x)\) \(\log(8^3) = 3\log(8)\)

Common 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:

  1. “\(\log_2(8) + \log_2(4) = \log_2(12)\)”

  2. “\(\frac{\log(100)}{\log(10)} = \log\left(\frac{100}{10}\right) = \log(10) = 1\)”

  3. “\(\log_3(9^x) = (\log_3(9))^x = 2^x\)”

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

  1. Evaluate: \(\log_4(64)\)

  2. Simplify: \(\log_3(9) + \log_3(27)\)

  3. Solve: \(\log_5(x + 4) = 2\)

  4. 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.

  1. Write down the equation for the value of the investment after \(t\) years.

  2. How long until the investment reaches 8,000 EUR?

  3. Use the Rule of 72 to estimate the doubling time, then compare it with the exact value from logarithms.

Think-Pair-Share: Which Offer Wins?

3 minutes alone, then 4 minutes with a partner

Two banks advertise savings plans:

  • Bank A: 6% interest per year for \(t\) years
  • Bank B: 3% interest per year, “but we give you twice the time!” (\(2t\) years)

Which deposit is worth more in the end? Justify your answer with exponents or logarithms - no calculator needed.

Compare the growth factors \((1.06)^t\) and \((1.03)^{2t} = \left((1.03)^2\right)^t\).

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

  1. Expand completely: \((x - 3)^3\)

Individual Exercise

Work individually for 5 minutes

  1. If \(\log_2(x) + \log_4(x) = 3\), find x.

  2. Expand: \((3x - 2y)^3\)

  3. Simplify: \(\log_3(27) - \log_3(3)\)

  4. 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

  1. Evaluate \(\log_5(125)\).

  2. Solve \(2^{3x} = 64\).

  3. Write \(2\log(x) + \log(5)\) as a single logarithm.

  4. 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!

Homework Assignment

Complete Tasks 01-05!