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74-hw9

# 74-hw9 - a,b has a smaller length than 1 2(How small can...

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MATH 74 HOMEWORK 9 - DUE MONDAY, APRIL 23 I have a function L : (0 , ) R satisfying the following properties: (1) L (10) = 1 (2) If 0 < x < y then L ( x ) < L ( y ) (3) For any x,y (0 , ) we have L ( xy ) = L ( x ) + L ( y ) Cite (1), (2), and (3) explicitly, by number, when you use them. 1. Prove that L (1) = 0. 2. Fix y (0 , ). Prove that L ( 1 y ) = - L ( y ). 3. Fix x (0 , ). Prove by induction that L ( x n ) = nL ( x ) for all n N . Problems 4 and 5. We can use the given information and the previous exercises to begin calculating other values of L . Suppose we wanted an idea of what L (30) is. We know L (30) > L (10) by (2) = 1 by (1) and we also know L (30 2 ) = L (900) < L (1000) by (2) = 3 L (10) by Exercise 3 with x = 10, n = 3 = 3 by (1) Since Exercise 3 with x = 30 and n = 2 tells us that L (30 2 ) = 2 L (30), we conclude 2 L (30) < 3 so that L (30) < 3 2 . All together, then, we have shown that whatever L (30) is it lies in the interval (1 , 3 2 ). 4. Use (1), (2), (3), and ideas similar to those shown, or any other ideas, to ﬁnd a number a 1 and a number b 3 2 and a proof that L (30) ( a,b ). I do not care which a and b you use provided that your interval (
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Unformatted text preview: a,b ) has a smaller length than 1 2 . (How small can you make it? Can you evaluate L (30) exactly?) 5. Use the same ideas to estimate L (50) (ie, ﬁnd a < b and prove L (50) ∈ ( a,b ), with b-a as small as you like; be as precise or as rough as you want). Remarks The base-10 logarithm function log x satisﬁes properties like (1), (2), (3). In precalculus it is likely that (1), (2), (3) (and their consequences, like Exercises 1-3) were all you knew about the logarithm. You might wonder: is there more than one function with these properties? (What would the others be? Why are they less important than log x ?) If there is only one function with these properties, we should be able to use (1), (2), (3) and nothing else to calculate logarithms to any degree of precision. (Does your experience with 4 and 5 suggest this is possible?)...
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