Logarithms

Grade 12 Math - Logarithms

Lesson Objectives

  • Review logarithmic laws and apply them in simplifying expressions.
  • Solve logarithmic equations with various bases.
  • Apply logarithms in real-world contexts including exponential growth/decay.
  • Use logarithmic tables and calculators for computation.

Lesson Introduction

Logarithms are powerful tools used in many scientific and mathematical computations. From scaling in scientific notation to solving exponential equations, logarithms allow us to transform complex multiplicative problems into simpler additive ones.

Core Lesson Content

Key Logarithmic Laws:

  • \log_b(MN) = \log_b M + \log_b N
  • \log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N
  • \log_b(M^k) = k \log_b M
  • \log_b b = 1 , \log_b 1 = 0
  • Change of base: \log_b x = \frac{\log_a x}{\log_a b}

Worked Example

Example 1: Simplify \log_{10} 1000 .
\log_{10} 1000 = \log_{10} (10^3) = 3 \log_{10} 10 = 3 \times 1 = 3
Example 2: Simplify \log_2 8 + \log_2 4 .
\log_2 8 = 3, \log_2 4 = 2 \Rightarrow 3 + 2 = 5
Example 3: Evaluate \log_5 25 - \log_5 5 .
\log_5 25 = 2, \log_5 5 = 1 \Rightarrow 2 - 1 = 1
Example 4: Solve for x in \log_3 x = 4 .
x = 3^4 = 81
Example 5: Solve \log_x 49 = 2 .
x^2 = 49 \Rightarrow x = 7
Example 6: Expand \log_2(16x^3) .
\log_2 16 + \log_2 x^3 = 4 + 3 \log_2 x
Example 7: Solve \log_{10}(2x - 1) = 1 .
2x - 1 = 10^1 = 10 \Rightarrow 2x = 11 \Rightarrow x = \frac{11}{2} = 5.5
Example 8: [WAEC] If \log_{10} x = 0.3010 , find x .
x = 10^{0.3010} \approx 2.0
Example 9: Use change of base to compute \log_3 81 .
\log_3 81 = \frac{\log_{10} 81}{\log_{10} 3} = \frac{1.9085}{0.4771} \approx 4
Example 10: Solve 2\log_2 x = 6 .
Divide: \log_2 x = 3 \Rightarrow x = 2^3 = 8

Exercises

  1. Simplify: \log_4 64
  2. [WAEC] Evaluate: \log_5 125 - \log_5 25 (Past Question)
  3. Expand: \log_3(27x^2)
  4. [NECO] Solve: \log_2 (x + 1) = 4 (Past Question)
  5. [WAEC] If \log_{10} x = 2.0000 , find x (Past Question)
  6. Use change of base to find \log_7 343
  7. Find x : \log_5 x = \log_5 20
  8. Solve: \log_2 (3x - 1) = 5
  9. [JAMB] If \log_{10} 2 = 0.3010 , find \log_{10} 8 (Past Question)
  10. Evaluate \frac{\log 1000}{\log 10}

Conclusion/Recap

Logarithms make solving exponential problems more manageable. Understanding the laws of logarithms helps simplify complex expressions and solve real-world problems like sound intensity, pH levels, and population growth. In the next lesson, we’ll explore logarithmic and exponential functions graphically.

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