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Mathematical Investigations III Name: Mathematical Investigations III Logarithms The Third Law of Logarithms Use a method similar to the ones that were used to prove the First and Second Laws of Logarithms to prove the Third Law of Logarithms: THE THIRD LAW OF LOGARITHMS: x log b = log b x log b y , b > 0, b 1 y Let = p, and =q Practice: 1. Rewrite the following using the Third Law of Logarithms and simplify if possible. a. b log b 5 b. Logs 6.1 7 log 2 8 Rev. S11 Mathematical Investigations III Name: 2. k Prove: log 7 log 7 k log 7 (11) . Use the method of proof used on page 6.1. 11 Putting It All Together Sometimes two or more laws of logarithms must be used in order to simplify an expression. Simplify each of the following using whatever laws you need. 3. 34 log 2 7 4 4. 3b12 log b 4 We will learn from subsequent sheets how to find a numerical value for an expression like #3 above. Because we don’t know the base of the logarithm in #4 we can’t find a value for such a problem. Often problems like this one are asked in a slightly different way (see page 6.3). Logs 6.2 Rev. S11 Mathematical Investigations III Name: Suppose that logb 2 p and logb 3 q . Now you can use the laws of logarithms to show that 8b 2 2 log b log b 8b log b 81 81 log b 8 log b b 2 log b 81 log b 23 log b b 2 log b 34 3log b 2 2 log b b 4 log b 3 3 p 2 4q Additional practice problems: Suppose that logb 2 p , logb 3 q , or logb 5 r . Give an expression in terms of p, q, or r for each of the following. 3. logb 6 4. log b 1024b 2 5. 25 log b 9 6. logb 7. log b 120b 7 8. log 2 b 9. Explain why logb 2 3 p q . Logs 6.3 3 400 (CAREFUL!) Rev. S11