Tutorial on Logarithms
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This document provides information about logarithmic and exponential forms including: - Examples of logarithmic expressions and their exponential equivalents - Rules for evaluating logarithmic expressions - Examples of using logarithms to solve equations - How to use logarithms to find the x-coordinate when given the y-coordinate on a curve
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Tutorial on Logarithms
- 1. Logarithms Prepared by Ismail Mohammad El-Badawy ismailelbadawy@gmail.com
- 2. Logarithmic and exponential forms • Log 2 8 • Log 3 81 • Log 6 36 2 to the power of ?? gives 8 3 to the power of ?? gives 81 6 to the power of ?? gives 36 What is the base 2 logarithm of 8 ? What is the base 3 logarithm of 81? What is the base 6 logarithm of 36?
- 3. Logarithmic and exponential forms • Log 2 8 = 3 23 = 8 • Log 3 81 = 4 34 = 81 • Log 6 36 = 2 62 = 36 2 to the power of 3 gives 8 3 to the power of 4 gives 81 6 to the power of 2 gives 36 What is the base 2 logarithm of 8 ? 3 What is the base 3 logarithm of 81? 4 What is the base 6 logarithm of 36? 2
- 4. Logarithmic and exponential forms Note: The default value of the base is 10. e.g. Log (100) = 2 102 = 100
- 5. • Find the value of a : • Log 4 𝑎 = 3 • Log 2 𝑎 = 5 • Log 5 𝑎 = 4 The argument is unknown
- 6. • Find the value of a : • Log 4 𝑎 = 3 ✓ 43 = 𝑎 ∴ 𝑎 = 64 • Log 2 𝑎 = 5 ✓ 25 = 𝑎 ∴ 𝑎 = 32 • Log 5 𝑎 = 4 ✓ 54 = 𝑎 ∴ 𝑎 = 625 The argument is unknown Check your answers 𝐿𝑜𝑔4 64 = 𝐿𝑜𝑔 64 𝐿𝑜𝑔 4 𝐿𝑜𝑔2 32 = 𝐿𝑜𝑔 32 𝐿𝑜𝑔 2 𝐿𝑜𝑔5 625 = 𝐿𝑜𝑔 625 𝐿𝑜𝑔 5
- 7. • Find the value of b : • Log 𝑏 81 = 2 • Log 𝑏 243 = 5 • Log 𝑏 216 = 3 The base is unknown
- 8. • Find the value of b : • Log 𝑏 81 = 2 ✓ 𝑏2 = 81 → 𝑏2 = 92 ∴ 𝑏 = 9 • Log 𝑏 243 = 5 ✓ 𝑏5 = 243 → 𝑏5 = 35 ∴ 𝑏 = 3 • Log 𝑏 216 = 3 ✓ 𝑏3 = 216 → 𝑏3 = 63 ∴ 𝑏 = 6 The base is unknown Check your answers 𝐿𝑜𝑔9 81 = 𝐿𝑜𝑔 81 𝐿𝑜𝑔 9 𝐿𝑜𝑔3 243 = 𝐿𝑜𝑔 243 𝐿𝑜𝑔 3 𝐿𝑜𝑔6 216 = 𝐿𝑜𝑔 216 𝐿𝑜𝑔 6
- 9. • Find the value of c : • Log 3 81 = 𝑐 • Log 7 49 = 𝑐 • Log 5 125 = 𝑐 The exponent is unknown
- 10. • Find the value of c : • Log 3 81 = 𝑐 ✓ 3 𝑐 = 81 → 3 𝑐 = 34 ∴ 𝑐 = 4 • Log 7 49 = 𝑐 ✓ 7 𝑐 = 49 → 7 𝑐 = 72 ∴ 𝑐 = 2 • Log 5 125 = 𝑐 ✓ 5 𝑐 = 125 → 5 𝑐 = 53 ∴ 𝑐 = 3 The exponent is unknown Check your answers 𝐿𝑜𝑔3 81 = 𝐿𝑜𝑔 81 𝐿𝑜𝑔 3 𝐿𝑜𝑔7 49 = 𝐿𝑜𝑔 49 𝐿𝑜𝑔 7 𝐿𝑜𝑔5 125 = 𝐿𝑜𝑔 125 𝐿𝑜𝑔 5
- 11. • Find the value of x : • Log 7 (2𝑥 − 5) = 3 • Log (𝑥2) = 4 • Log 5 3𝑥 = 1.6
- 12. • Find the value of x : • Log 7 (2𝑥 − 5) = 3 ✓ 73 = 2𝑥 − 5 → 343 = 2𝑥 − 5 ∴ 𝑥 = 343 + 5 ÷ 2 = 174 • Log (𝑥2) = 4 ✓ 104 = 𝑥2 → 10000 = 𝑥2 ∴ 𝑥 = ± 10000 = ±100 • Log 5 3𝑥 = 1.6 ✓ 51.6 = 3𝑥 → 13.1 = 3𝑥 ∴ 𝑥 = 13.1 ÷ 3 = 4.37
- 13. • Find the value of x : • Log (𝑥−3) 5 = 0.5 • Log 2 16 = 7𝑥 − 3
- 14. • Find the value of x : • Log (𝑥−3) 5 = 0.5 ✓ (𝑥 − 3)0.5 = 5 → 𝑥 − 3 = 5 → 𝑥 − 3 = 52 → 𝑥 − 3 = 25 ∴ 𝑥 = 28 • Log 2 16 = 7𝑥 − 3 ✓ 2(7𝑥−3) = 16 → 2(7𝑥−3) = 24 → 7𝑥 − 3 = 4 ∴ 𝑥 = 1
- 15. • Log 4 4 = ? ? • Log 3 1 = ? ? • Log 6 ? ? = 1 • Log 5 ? ? = 0 • Log 2 12 = Log 2 3 + Log 2 ? ? • Log 𝑥 (? ? ) = Log 𝑥 5 + Log 𝑥 𝑦 Rules of logs
- 16. • Log 4 4 = 1 • Log 3 1 = 0 • Log 6 6 = 1 • Log 5 1 = 0 • Log 2 12 = Log 2 3 + Log 2 4 • Log 𝑥 (5𝑦) = Log 𝑥 5 + Log 𝑥 𝑦 Rules of logs
- 17. • Log 3 25 = ? ? Log 3 5 • 4 Log 7 2 = Log 7 (? ? ) • Log 2 12 = Log 2 36 − Log 2 ? ? • Log ?? ?? = Log 𝑥 − Log 5𝑦 • Log 4 2 10 = −Log 4 (? ? ) • Log 3 25 = Log 𝑥 (??) Log 𝑥 (??) Rules of logs
- 18. • Log 3 25 = 2 Log 3 5 • 4 Log 7 2 = Log 7 (16) • Log 2 12 = Log 2 36 − Log 2 3 • Log 𝑥 5𝑦 = Log 𝑥 − Log 5𝑦 • Log 4 2 10 = −Log 4 (5) • Log 3 25 = Log 𝑥 (25) Log 𝑥 (3) Rules of logs
- 19. • Evaluate: • Log 2 1 16 • Log 16 8 = • Log 5 1 Using rules of Logs
- 20. • Evaluate: • Log 2 1 16 ✓ Log 2 1 16 = −Log 2 16 = −4 • Log 16 8 ✓ Log 16 8 = Log 2 8 Log 2 16 = 3 4 • Log 5 1 = 0 Using rules of Logs
- 21. • Evaluate: • Log 5 3 − Log 5 75 • 2 Log 2 6 − Log 2 9 • 1 2 Log 10 4 + Log 10 50 Using rules of Logs
- 22. • Evaluate: • Log 5 3 − Log 5 75 ✓ Log 5 3 75 = Log 5 1 25 = −Log 5 25 = −2 • 2 Log 2 6 − Log 2 9 ✓ Log 2 62 − Log 2 9 = Log 2 36 − Log 2 9 = Log 2 36 9 = Log 2 4 = 2 • 1 2 Log 10 4 + Log 10 50 ✓ Log 10 40.5 + Log 10 50 = Log 10 2 + Log 10 50 = Log 10 2 × 50 = Log 10 100 = 2 Using rules of Logs
- 23. • Simplify the following logarithms: • Log 10 + Log 2 − Log 4 • 1 2 Log 𝑥 − 3 Log y • 1 3 Log 8 + 2 Log 4 Using rules of Logs
- 24. • Simplify the following logarithms: • Log 10 + Log 2 − Log 4 ✓ Log 10 × 2 − Log 4 = Log 20 − Log 4 = Log 20 4 = Log 5 • 1 2 Log 𝑥 − 3 Log y ✓ Log 𝑥 Τ1 2 − Log 𝑦3 = Log 𝑥 𝑦3 • 1 3 Log 8 + 2 Log 4 = Log 8 Τ1 3 + Log 42 = Log 2 + Log 16 = Log 2 × 16 = Log 32 Using rules of Logs
- 25. • The point P on the curve 𝑦 = 9 𝑥 has y-coordinate equal to 150. Use logarithms to find the x-coordinate of P, correct to 3 significant figures. Using rules of Logs
- 26. • The point P on the curve 𝑦 = 9 𝑥 has y-coordinate equal to 150. Use logarithms to find the x-coordinate of P, correct to 3 significant figures. ✓ 150 = 9 𝑥 Log 150 = Log 9 𝑥 Log 150 = 𝑥 Log 9 ∴ 𝑥 = Log 150 Log 9 = Log9150 = 2.28 Using rules of Logs
- 27. • Given that Log 𝑥 5𝑦 + 1 − Log 𝑥 3 = 4, express y in terms of x. Using rules of Logs
- 28. • Given that Log 𝑥 5𝑦 + 1 − Log 𝑥 3 = 4, express y in terms of x. ✓ Log 𝑥 5𝑦+1 3 = 4 𝑥4 = 5𝑦 + 1 3 3𝑥4 = 5𝑦 + 1 ∴ 𝑦 = 3𝑥4 − 1 5 Using rules of Logs
- 29. • Use logarithms to solve the following equation, giving the value of x correct to 3 s.f: • 7 𝑥 = 2 𝑥+1 Using rules of Logs
- 30. • Use logarithms to solve the following equation, giving the value of x correct to 3 s.f: • 7 𝑥 = 2 𝑥+1 ✓ Log 7 𝑥 = Log 2 𝑥+1 → 𝑥 Log 7 = 𝑥 + 1 Log 2 𝑥 𝑥+1 = Log 2 Log 7 = 0.356 𝑥 = 𝑥 + 1 0.356 = 0.356𝑥 + 0.356 𝑥 − 0.356𝑥 = 0.356 0.644 𝑥 = 0.356 ∴ 𝑥 = 0.356 ÷ 0.644 = 0.553 Using rules of Logs
- 31. Assignment