# Lect6 - Logarithmic Functions Definition: Let a > 0, a 1....

This preview shows pages 1–4. Sign up to view the full content.

92.131 Lecture 6 1 of 16 Ronald Brent © 2009 All rights reserved. Logarithmic Functions Definition: Let a > 0, 1 a . Then x a log is the number to which you raise a to get x. Logarithms are in essence exponents. Their domains are powers of the base and their ranges are the exponents needed to produce those particular powers. Example: Demonstrate that 2 16 log 4 = . Here the base is 4 and x = 16. To what number do you have to raise 4 in order to get 16? Answer: 2, so 2 16 log 4 = . Example: Show that 4 000 , 10 log 10 = . Here the base is 10, and x = 10,000. What number do you have to raise 10 to, in order to get 10,000 (4 zeros)? Answer: 4, so 4 000 , 10 log 10 = .

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

View Full Document
92.131 Lecture 6 2 of 16 Ronald Brent © 2009 All rights reserved. Example: ? 001 . log 10 = Here the base is 10, and x = .001 . Write .001 as a power of 10. Since 3 10 001 . = , 3 001 . log 10 = . The function x 10 log is known as the common logarithm. Sometimes you will see the expression x log . Theorem: Let a > 0, 1 a . Then x a log and x a are inverse to each other. Remark: This theorem is too difficult to prove here. However, if you let x a x f = ) ( and x x g a log ) ( = , you can show the following.
92.131 Lecture 6 3 of 16 Ronald Brent © 2009 All rights reserved. a) () x x g a a a x g f log ) ( ) ( = = . What does this mean? It is a , raised to the number to which you raise a to get x . So it equals x , that is x a a x g f x x g a = = = log ) ( ) (.

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.

## This note was uploaded on 02/13/2012 for the course MATH 92.131 taught by Professor Staff during the Fall '09 term at UMass Lowell.

### Page1 / 16

Lect6 - Logarithmic Functions Definition: Let a > 0, a 1....

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online