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Unformatted text preview: Math021, week 3 Remark 3.1 log a x is NOT defined when x ≤ . In other words, the domain of log a is the collection of all positive numbers. Proposition 3.2 For any numbers x , y , a , b > and for any number k , 1. log a xy = log a x + log a y . 2. log a x y = log a x- log a y . 3. log a x k = b log a x . 4. log a x = log b x log b a proof: Let log a x = u , log a y = v , proof of 1): Since a u + v = a u a v = xy, we have log a xy = u + v = log a x + log a y. proof of 2): log a x = log a ( x y y ) = log a x y + log a y. proof of 3): Since a bu = ( a u ) b = x b , we have log a x b = bu = b log a x. proof of 4): Since u log b a = log b a u = log b x, we have log a x = u = log b x log b a . Example 3.3 A certain election committee consists of 400 people initially. The committee doubles its size and vote for a president once every four years. How long we have to wait until the committee reaches a size of 6000000 ? 1 solution: The size of the given committee is 400(2) t after 4 t years (see the previous example on carbon dating). Suppose that we need to wait for the T th election (that is, after 4 T years) so that the size of the committee is 6000000. Then, 400(2) T = 6000000 ....
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This note was uploaded on 02/13/2012 for the course MATH 021 taught by Professor Luxnu during the Fall '10 term at HKUST.
- Fall '10