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

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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 = .
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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.
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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 ) ( ) (.
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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.

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

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