Chapter 4: Indices, Surds and Logarithms

Overview

Indices, surds, and logarithms are fundamental algebraic concepts that extend our understanding of exponentiation and provide powerful tools for simplifying complex expressions. This chapter explores the laws of indices, properties of surds, and the fascinating world of logarithms. These concepts are essential for solving exponential equations, understanding growth and decay patterns, and forming the foundation for advanced mathematical analysis.

Learning Objectives

After completing this chapter, you will be able to:

• Apply laws of indices to simplify exponential expressions
• Simplify and manipulate surds
• Rationalize denominators containing surds
• Use logarithmic laws to simplify expressions
• Solve exponential and logarithmic equations
• Apply these concepts in real-world scenarios

Key Concepts

4.1 Laws of Indices

Definition of Indices

An index (or exponent) is the power to which a base number is raised. In $a^m$, '$a$' is the base and '$m$' is the index.

Fundamental Laws of Indices

1. Product of Powers: $a^m \times a^n = a^{m+n}$
2. Quotient of Powers: $a^m \div a^n = a^{m-n}$
3. Power of a Power: $(a^m)^n = a^{mn}$
4. Power of a Product: $(ab)^n = a^n b^n$
5. Zero Index: $a^0 = 1$ (where $a \neq 0$)
6. Negative Index: $a^{-n} = \frac{1}{a^n}$
7. Fractional Index: $a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$

4.2 Laws of Surds

Definition of Surds

A surd is an irrational number that can be expressed as the root of a rational number. Examples include $\sqrt{2}$, $\sqrt[3]{5}$, and $\sqrt{\frac{1}{3}}$.

Properties of Surds

1. Product of Surds: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$
2. Quotient of Surds: $\sqrt{a} \div \sqrt{b} = \sqrt{\frac{a}{b}}$
3. Combining Like Surds: $m\sqrt{a} \pm n\sqrt{a} = (m \pm n)\sqrt{a}$

Rationalizing Denominators

Conjugate Surd Method: The conjugate of $(\sqrt{a} + \sqrt{b})$ is $(\sqrt{a} - \sqrt{b})$. Multiplying them gives: $(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b$ (a rational number)

4.3 Laws of Logarithms

Definition of Logarithms

A logarithm is the inverse operation of exponentiation. If $a^x = N$, then $\log_a N = x$.

Common Logarithms:

• $\log_{10} x$ or $\lg x$ (common logarithm)
• $\log_e x$ or $\ln x$ (natural logarithm)

Fundamental Laws of Logarithms

1. Product Law: $\log_a(mn) = \log_a m + \log_a n$
2. Quotient Law: $\log_a\left(\frac{m}{n}\right) = \log_a m - \log_a n$
3. Power Law: $\log_a(m^p) = p \log_a m$
4. Basic Properties: $\log_a a = 1$ and $\log_a 1 = 0$

Change of Base Formula

$$\log_a b = \frac{\log_c b}{\log_c a}$$

Important Formulas and Methods

Exponent Simplification

Key Strategies:

• Apply laws of indices systematically
• Convert negative and fractional exponents to radical form
• Simplify expressions step by step

Surd Manipulation

Rationalization Techniques:

1. Single surd denominator: Multiply numerator and denominator by the surd
2. Binomial surd denominator: Use conjugate method

Logarithmic Problem Solving

Key Strategies:

• Convert between exponential and logarithmic forms
• Apply logarithmic laws to simplify complex expressions
• Use change of base for calculator computations

Solved Examples

Example 1: Simplifying Indices

Simplify the following expressions: a) $3^4 \times 3^{-2}$ b) $(2^3)^{2/3}$ c) $16^{-3/4}$ d) $(a^2b^3) \times (a^{-1}b^2)$

Solutions:

a) $3^4 \times 3^{-2} = 3^{4-2} = 3^2 = 9$

b) $(2^3)^{2/3} = 2^{3 \times \frac{2}{3}} = 2^2 = 4$

c) $16^{-3/4} = (2^4)^{-3/4} = 2^{-3} = \frac{1}{8}$

d) $(a^2b^3) \times (a^{-1}b^2) = a^{2-1} \times b^{3+2} = a b^5$

Example 2: Surd Simplification

Simplify: a) $\sqrt{12} + \sqrt{27} - \sqrt{48}$ b) $\sqrt{8} \times \sqrt{18}$ c) $(3 + \sqrt{5})(3 - \sqrt{5})$ d) $\frac{2}{\sqrt{3} + \sqrt{2}}$

Solutions:

a) $\sqrt{12} + \sqrt{27} - \sqrt{48} = 2\sqrt{3} + 3\sqrt{3} - 4\sqrt{3} = (2 + 3 - 4)\sqrt{3} = \sqrt{3}$

b) $\sqrt{8} \times \sqrt{18} = \sqrt{8 \times 18} = \sqrt{144} = 12$

c) $(3 + \sqrt{5})(3 - \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4$

d) $\frac{2}{\sqrt{3} + \sqrt{2}} = \frac{2(\sqrt{3} - \sqrt{2})}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})} = \frac{2(\sqrt{3} - \sqrt{2})}{3 - 2} = 2(\sqrt{3} - \sqrt{2})$

Example 3: Logarithmic Calculations

Evaluate: a) $\log_2 8 + \log_2 32$ b) $\log_3 27 - \log_3 3$ c) $\log_5(25^2)$ d) $\log_2 10 + \log_3 10$

Solutions:

a) $\log_2 8 + \log_2 32 = \log_2(8 \times 32) = \log_2 256 = 8$

b) $\log_3 27 - \log_3 3 = \log_3(\frac{27}{3}) = \log_3 9 = 2$

c) $\log_5(25^2) = 2 \log_5 25 = 2 \times 2 = 4$

d) $\log_2 10 + \log_3 10 = \log_2 10 + \log_3 10$ Cannot be simplified further with basic laws

Example 4: Solving Exponential Equations

Solve: a) $2^{x+1} = 8$ b) $3^{2x} = 81$ c) $4^x = 2^{x+3}$ d) $2^{x^2} = 16^x$

Solutions:

a) $2^{x+1} = 8 = 2^3$ $x + 1 = 3 \Rightarrow x = 2$

b) $3^{2x} = 81 = 3^4$ $2x = 4 \Rightarrow x = 2$

c) $4^x = 2^{x+3}$ $(2^2)^x = 2^{x+3}$ $2^{2x} = 2^{x+3}$ $2x = x + 3 \Rightarrow x = 3$

d) $2^{x^2} = 16^x = (2^4)^x = 2^{4x}$ $x^2 = 4x$ $x^2 - 4x = 0$ $x(x - 4) = 0$ $x = 0$ or $x = 4$

Example 5: Solving Logarithmic Equations

Solve: a) $\log_3(x+2) = 2$ b) $\log_4 x + \log_4 4 = 3$ c) $2 \log_5 x = \log_5 36$ d) $\log_2(x-1) + \log_2(x+1) = 3$

Solutions:

a) $\log_3(x+2) = 2$ $x + 2 = 3^2 = 9$ $x = 7$

b) $\log_4 x + \log_4 4 = 3$ $\log_4(4x) = 3$ $4x = 4^3 = 64$ $x = 16$

c) $2 \log_5 x = \log_5 36$ $\log_5(x^2) = \log_5 36$ $x^2 = 36$ $x = 6$ or $x = -6$ But $x > 0$ for $\log_5 x$ to be defined, so $x = 6$

d) $\log_2(x-1) + \log_2(x+1) = 3$ $\log_2[(x-1)(x+1)] = 3$ $\log_2(x^2 - 1) = 3$ $x^2 - 1 = 2^3 = 8$ $x^2 = 9$ $x = 3$ or $x = -3$ But $x > 1$ for logs to be defined, so $x = 3$

Mathematical Derivations

Proof of Logarithm Laws

Product Law Proof: Let $\log_a m = x$ and $\log_a n = y$ Then $a^x = m$ and $a^y = n$ $mn = a^x \times a^y = a^{x+y}$ Therefore, $\log_a(mn) = x + y = \log_a m + \log_a n$

Power Law Proof: Let $\log_a m = x$ Then $a^x = m$ $m^p = (a^x)^p = a^{xp}$ Therefore, $\log_a(m^p) = xp = p \log_a m$

Change of Base Formula Derivation

Let $\log_a b = x$ Then $a^x = b$ Taking $\log_c$ of both sides: $\log_c(a^x) = \log_c b$ $x \log_c a = \log_c b$ $x = \frac{\log_c b}{\log_c a}$ Therefore, $\log_a b = \frac{\log_c b}{\log_c a}$

Real-World Applications

1. Population Growth

Exponential growth models: $P(t) = P_0 e^{rt}$

Example: A population grows at 3% annually. Initial population 10,000. Find population after 10 years.

$P(10) = 10000 e^{0.03 \times 10} = 10000 e^{0.3} \approx 13,498$ people

2. Radioactive Decay

Half-life formula: $N(t) = N_0 (\frac{1}{2})^{t/T}$

Example: A radioactive substance has half-life of 5 years. Initial amount 100g. Find amount remaining after 15 years.

$N(15) = 100 \times (\frac{1}{2})^{15/5} = 100 \times (\frac{1}{2})^3 = 100 \times \frac{1}{8} = 12.5$g

3. pH Calculations

$pH = -\log_{10}[H^+]$

Example: Find pH of solution with $[H^+] = 10^{-7}$ M $pH = -\log_{10}(10^{-7}) = 7$

4. Richter Scale for Earthquakes

Magnitude = $\log_{10}(\frac{I}{I_0})$

Example: Earthquake with intensity 1000 times reference magnitude Magnitude = $\log_{10}(1000) = 3$

Complex Problem-Solving Techniques

Problem: Simplify $a^{\log_a b \times \log_b c \times \log_c a}$

Solution: Let $x = \log_a b$, $y = \log_b c$, $z = \log_c a$ Then $a^x = b$, $b^y = c$, $c^z = a$ Substitute: $c = b^y = (a^x)^y = a^{xy}$ Then $a = c^z = (a^{xy})^z = a^{xyz}$ Therefore, $xyz = 1$ So the expression simplifies to $a^1 = a$

Problem: Solve $2^{2x} - 6 \times 2^x + 8 = 0$

Solution: Let $y = 2^x$ Then equation becomes: $y^2 - 6y + 8 = 0$ $(y - 2)(y - 4) = 0$ $y = 2$ or $y = 4$ So $2^x = 2 \Rightarrow x = 1$ Or $2^x = 4 \Rightarrow x = 2$

Problem: If $\log_{12} 27 = a$, express $\log_6 16$ in terms of $a$

Solution: Using change of base formula: $\log_{12} 27 = \log_{12}(3^3) = 3 \log_{12} 3 = a \Rightarrow \log_{12} 3 = \frac{a}{3}$

Now, $\log_6 16 = \log_6(2^4) = 4 \log_6 2$ $= \frac{4 \log_{12} 2}{\log_{12} 6}$ (change of base) $= \frac{4 \log_{12} 2}{\log_{12}(2 \times 3)}$ $= \frac{4 \log_{12} 2}{\log_{12} 2 + \log_{12} 3}$

We need to find $\log_{12} 2$: Note that $\log_{12} 3 + \log_{12} 4 = \log_{12} 12 = 1$ $\log_{12} 3 + 2 \log_{12} 2 = 1$ $\frac{a}{3} + 2 \log_{12} 2 = 1$ $2 \log_{12} 2 = 1 - \frac{a}{3} = \frac{3 - a}{3}$ $\log_{12} 2 = \frac{3 - a}{6}$

Therefore: $\log_6 16 = \frac{4 \times \frac{3 - a}{6}}{\frac{3 - a}{6} + \frac{a}{3}}$ $= \frac{4 \times \frac{3 - a}{6}}{\frac{3 - a + 2a}{6}}$ $= \frac{4 \times \frac{3 - a}{6}}{\frac{3 + a}{6}}$ $= \frac{4(3 - a)}{3 + a}$

Summary Points

• Indices represent repeated multiplication and follow specific laws
• Surds are irrational roots that can be simplified using properties
• Logarithms are the inverse of exponentiation and solve exponential equations
• All three concepts are interconnected and used in various real-world applications
• Master these techniques for success in higher mathematics

Common Mistakes to Avoid

1. Index law confusion - Remember the laws correctly (product adds exponents)
2. Surd simplification - Always simplify to simplest radical form
3. Logarithm domain - Arguments must be positive
4. Change of base - Apply the formula correctly
5. Rationalization - Use appropriate conjugates for denominators

SPM Exam Tips

Exam Strategies

1. Memorize laws thoroughly - Index, surd, and logarithm laws
2. Practice simplification - Work with complex expressions regularly
3. Equation solving - Master both exponential and logarithmic equations
4. Word problems - Translate real situations to mathematical expressions
5. Calculator skills - Use change of base for complex logarithms

Key Exam Topics

• Indices simplification (25% of questions)
• Surd manipulation (20% of questions)
• Logarithmic laws (25% of questions)
• Exponential equations (20% of questions)
• Logarithmic equations (10% of questions)

Time Management Tips

• Basic simplification: 2-3 minutes
• Surd rationalization: 3-4 minutes
• Exponential equations: 4-5 minutes
• Logarithmic equations: 5-6 minutes
• Complex applications: 7-8 minutes

Practice Problems

Level 1: Indices

1. Simplify: a) $5^3 \times 5^{-2}$ b) $(2^4)^{3/2}$ c) $81^{-1/4}$ d) $(x^3y^2) \times (x^{-1}y^4)$

2. Evaluate without calculator: a) $4^{1/2} + 8^{1/3}$ b) $27^{-2/3}$ c) $16^{3/4}$

Level 2: Surds

1. Simplify: a) $\sqrt{50} + \sqrt{18} - \sqrt{8}$ b) $\sqrt{12} \times \sqrt{27}$ c) $(5 - \sqrt{3})(5 + \sqrt{3})$ d) $\frac{1}{2 + \sqrt{3}}$

2. Rationalize denominators: a) $\frac{3}{\sqrt{5} - \sqrt{2}}$ b) $\frac{2}{\sqrt{7} + 3}$ c) $\frac{1}{3\sqrt{2} - 2\sqrt{3}}$

Level 3: Logarithms

1. Evaluate: a) $\log_3 27 + \log_3 3$ b) $\log_2 64 - \log_2 4$ c) $\log_5(125^2)$ d) $\log_4 8 + \log_2 16$

2. Solve equations: a) $\log_7(x-1) = 2$ b) $\log_3 x + \log_3 9 = 4$ c) $3 \log_2 x = \log_2 243$ d) $\log(x+2) + \log(x-2) = 1$ (base 10)

Level 4: Applications

1. Investment: RM5000 invested at 8% annual compound interest. Find amount after 10 years.

2. Radioactive Decay: Substance with half-life 20 years. Initial mass 200g. Find mass after 60 years.

3. Earthquake: Earthquake magnitude 5.2 on Richter scale. Find intensity relative to reference.

4. pH: Solution has $[H^+] = 2.5 \times 10^{-4}$ M. Find pH.

Did You Know? 📚

John Napier, a Scottish mathematician, invented logarithms in 1614 as a tool to simplify complex calculations. Before calculators, logarithm tables were essential for astronomers, navigators, and engineers, reducing multiplication and division to addition and subtraction. The word "logarithm" comes from Greek: "logos" (ratio) and "arithmos" (number).

Quick Reference Guide

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Indices, surds, and logarithms provide powerful tools for simplifying complex expressions and solving real-world problems. These concepts are essential for understanding exponential growth, decay, and logarithmic scales used in various scientific applications.