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Mathematical Investigations III Name: Mathematical Investigations III Logarithms The Second Law of Logarithms On the same coordinate axes, graph the following functions using an appropriate window. All are base 10. f1 (x) log x 3 f2 ( x) log( 3) log( x) f3 (x) 3log x f 4 ( x) log x 3 Describe the graphs. Which, if any, are the same? The two identical graphs give us a result that we will call the Second Law of Logarithms. THE SECOND LAW OF LOGARITHMS: log b x k k log b x , b 0, b 1 We must prove this. As before, we begin by naming the log b x . Let log b x p . Change this into exponential form. Since we are interested in x k , raise both sides of your equation to the kth power and use a known property of exponents to simplify. Finish the proof. (Note: Two quick steps remain. Look back at the last proof if necessary.) Logs 5.1 Rev. S03 Mathematical Investigations III Name: Practice: 1. Rewrite the following using the Second Law of Logarithms and simplify if possible. 2. b. log2 35 = d. log3 x 4 = a. logb b7 c. log3 95 e. log 49 73x Prove: If m > 0, then log5 m3n 3nlog5 m . Use the method of proof used on page 5.1. Start by choosing what expressions to define as p and q. Logs 5.2 Rev. S03