note3 - Math021, week 3 Remark 3.1 log a x is NOT defined...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

This note was uploaded on 02/13/2012 for the course MATH 021 taught by Professor Luxnu during the Fall '10 term at HKUST.

Page1 / 5

note3 - Math021, week 3 Remark 3.1 log a x is NOT defined...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online