Topic: Indices, Indicial Equations, Logarithm and their Applications

Lesson Objectives

By the end of the lesson, students should be able to: - Define and use the laws of indices. - Solve simple problems involving indicial equations. - Define logarithms and state the laws of logarithms. - Apply logarithms to solve numerical problems. - Apply logarithms in real-life problems (e.g., compound interest, exponential growth/decay).

Previous Knowledge

Students are already familiar with basic algebra, simple equations, and powers of numbers.

Instructional Materials

  • Whiteboard/Marker or Chalkboard/Chalk
  • Graph paper
  • Scientific calculator
  • Charts showing laws of indices and logarithms

Introduction

Begin by asking students:
“What is \[2^3? What is 5^0? What is 1 / 2^{-3}\]?”
This leads to the concept of indices and their rules.

Teaching Content

1. Indices

Definition: If \(a\) is a real number and \(n\) is an integer, then
\[ a^n = a \times a \times \ldots \times a \quad (n \text{ times}) \]

Laws of Indices: - \(a^m \times a^n = a^{m+n}\) - \(\frac{a^m}{a^n} = a^{m-n}, \; a \neq 0\) - \((a^m)^n = a^{mn}\) - \(a^0 = 1, \; a \neq 0\) - \(a^{-n} = \frac{1}{a^n}\)

Example 1: Simplify \(2^3 \times 2^4\).
Solution: \(2^{3+4} = 2^7 = 128\).

Example 2: Solve \(\frac{5^6}{5^2}\).
Solution: \(5^{6-2} = 5^4 = 625\).

2. Indicial Equations

Definition: Equations in which the unknown appears as an index (power).

Example 1: Solve \(2^x = 16\).
Solution: \(2^x = 2^4 \implies x = 4\).

Example 2: Solve \(9^{x+1} = 81\).
Solution: \(9^{x+1} = 9^2 \implies x+1 = 2 \implies x = 1\).

3. Logarithms

Definition: If \(a^x = N\), then
\[ \log_a N = x \]
where \(a\) is the base, \(N > 0\), \(a > 0, a \neq 1\).

Laws of Logarithms: - \(\log_a (MN) = \log_a M + \log_a N\) - \(\log_a (M/N) = \log_a M - \log_a N\) - \(\log_a (M^k) = k \log_a M\) - \(\log_a a = 1, \; \log_a 1 = 0\)

Example 1: Solve \(\log_2 8\).
Solution: Since \(2^3 = 8\), \(\log_2 8 = 3\).

Example 2: Evaluate \(\log_{10} 1000\).
Solution: Since \(10^3 = 1000\), answer is 3.

4. Applications of Logarithms

Example 1 (Multiplication):
Evaluate \(234.5 \times 68.9\) using logarithm tables.

Example 2 (Exponential Growth):
The population of a town grows at 5% per year. If the initial population is 20,000, find the population after 3 years.

Formula:
\[ P = P_0 (1+r)^t \]

Substitution:
\[ P = 20000 (1.05)^3 = 23152.5 \]

Graphical Illustration

Class Activity

  1. Simplify: \((2^3 \times 2^5) \div 2^4\).
  2. Solve: \(3^x = 81\).
  3. Evaluate: \(\log_5 25\).
  4. Using logarithms, find the value of \(42.5 \times 0.00367\).

Evaluation Questions

  • State two laws of indices and give examples.
  • Solve: \(4^{2x} = 64\).
  • Find \(x\) if \(\log_{10} x = 2.5\).
  • State two applications of logarithms in daily life.

Conclusion

  • Indices deal with powers of numbers and have clear rules.
  • Indicial equations require expressing both sides as powers of the same base.
  • Logarithms are the inverse of indices.
  • Logarithms simplify multiplication, division, powers, and exponential problems.

Assignment

  1. Solve: \(5^{2x-1} = 125\).
  2. Evaluate: \(\log_3 81\).
  3. A population of 10,000 increases by 8% annually. Find the population after 2 years.