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📚 MATHSKILLER TEXTBOOK SERIES
03

Indices & Logarithms

Laws, Properties & Applications
© 2025 MathsKiller | mathskiller.pro
CHAPTER 3

Indices & Logarithms

🎯 Learning Objectives

3.1
Laws of Indices
LawFormulaExample
Multiplication$a^m \times a^n = a^{m+n}$$2^3 \times 2^4 = 2^7$
Division$a^m \div a^n = a^{m-n}$$3^5 \div 3^2 = 3^3$
Power of Power$(a^m)^n = a^{mn}$$(2^3)^4 = 2^{12}$
Zero Power$a^0 = 1$$5^0 = 1$
Negative Index$a^{-n} = \dfrac{1}{a^n}$$2^{-3} = \dfrac{1}{8}$
Fractional Index$a^{\frac{m}{n}} = \sqrt[n]{a^m}$$8^{\frac{2}{3}} = 4$
3.2
Definition of Logarithms
Logarithm ↔ Index Conversion
$\log_a N = x \Longleftrightarrow a^x = N$

⚡ Key Values to Remember

• $\log_a 1 = 0$ (because $a^0 = 1$)

• $\log_a a = 1$ (because $a^1 = a$)

3.3
Laws of Logarithms
LawFormula
Product$\log_a(MN) = \log_a M + \log_a N$
Quotient$\log_a\left(\dfrac{M}{N}\right) = \log_a M - \log_a N$
Power$\log_a M^n = n\log_a M$
3.4
Change of Base Formula
$\log_a b = \dfrac{\log_c b}{\log_c a} = \dfrac{\lg b}{\lg a}$

⚡ Calculator Trick

To find $\log_2 5$: Enter log 5 ÷ log 2 =

Answer: $\approx 2.322$

📋 DSE Past Paper Questions

DSE 2020 Q7

Simplify $\dfrac{(8^n)^2}{2^{5n}}$.

📝 Solution

$= \dfrac{(2^3)^{2n}}{2^{5n}} = \dfrac{2^{6n}}{2^{5n}} = 2^n$

DSE 2023 Q15

If $5^x = 3$, find the value of $25^x$.

A. 6   B. 9   C. 15   D. 243

📝 Quick Solution

$25^x = (5^2)^x = (5^x)^2 = 3^2 = 9$

Answer: B

📝 Practice Questions

1. Simplify: $\dfrac{3^{n+2} \times 9^n}{27^n}$
2. Find the value of $\log_2 32$.
3. If $\log_5 x = 3$, find $x$.
4. Simplify: $\log_3 9 + \log_3 27$
5. Solve: $2^{x+1} = 16$

📋 Answers

1. $9$   2. $5$   3. $125$   4. $5$   5. $x = 3$

📘 MathsKiller Textbook Series | Chapter 3: Indices & Logarithms

© 2025 MathsKiller | mathskiller.pro