Functions+Notes+_updated_.pdf

This is read the base 2 logarithm of 8 is 3 or the

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This is read “the base 2 logarithm of 8 is 3” or “the log, base 2, of 8 is 3” Here is the general definition: The base b logarithm of x , log b x , is the power to which we need to raise b in order to get x . Symbolically, log b x = y, means b y = x Logarithmic form Exponential form
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24 J. S´ anchez-Ortega Remark 2.11. The number log b x is defined only if b and x are both positive and b 6 = 1. Thus, it is not possible to compute, for example, log 3 ( - 9) (because there is no power of 3 that equals to - 9) and log 1 (2) (because there is no power of 1 that equals to 2). Some calculators do not permit direct calculation of logarithms other than
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2. Functions 25 common and natural logarithms. To compute logarithms with such calcula- tors, we can use the following Change-of-Base Formula : log b a = log a log b = ln a ln b Example 2.12. log 11 9 = log 9 log 11 0 . 91631 Some of the important algebraic properties of logarithms are listed below: Logarithm Identities. Relationship with Exponential Functions Let a and b positive numbers distinct from 1. Then for all positive numbers x and y , and every real number r , the following identities hold. Notice that they are consequences of the laws of exponents. 1. log b ( xy ) = log b x + log b y 2. log b x y = log b x - log b y 3. log b ( x r ) = r log b x 4. log b b = 1; log b 1 = 0 5. log b 1 y = - log b x 6. log b x = log a x log a b 7. log b ( b x ) = x The power to which you raise b in order to get b x is x ln( e x ) = x 8. b log b x = x Raising b to the power to which it must be raised to get x , yields x e ln x = x
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26 J. S´ anchez-Ortega Examples 2.13. log 2 4 = log 2 2 + log 2 2 = 1 + 1 = 2 , log 2 1 4 = - log 2 4 = - 2 log 2 (2 7 ) = 7 , 5 log 5 8 = 8 Definition 2.14. A logarithmic function has the form f ( x ) = log b x + C, ( b and C are constants with b > 0, b 6 = 1) , or, alternatively, f ( x ) = A ln x + C, ( A and C are constants with A 6 = 0) . Remark 2.15. f ( x ) = log b x + C = ln x ln b + C = 1 ln b ln x + C = A ln x + C, for A = 1 / ln b . Example 2.16. Sketch the graph of f ( x ) = log 2 x by hand. Solution: To do so, we determine a table of values. Graphing the points and joining them we get the graph on your left. On your right you could see the graphs of log b x for b = 1 / 4 , 1 / 2, and 4.
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2. Functions 27 Using the Laws of Logarithms Example 2.17. Use the logarithm identities to find the exact value of each expression: 1. log 2 2 3 64 32 8 2. log 2 6 - log 2 15 + log 2 20 Solution: 1. Apply log identity 3 to get log 2 2 3 64 32 8 ! = log 2 2 3 · 2 3 2 5 · 2 3 2 = log 2 2 6 2 13 2 = log 2 2 - 1 2 = - 1 2 log 2 2 = = - 1 2 2. Using log identities 1 and 2 we obtain log 2 6 - log 2 15+log 2 20 = log 2 (6 · 20) - log 2 15 = log 2 120 15 = log 2 8 = 3 Example 2.18. Use the logarithm identities to express the following quantities as a single logarithm: 1. ln a + 1 2 ln b 2. ln( a + b ) + ln( a - b ) - 2 ln c 3. ln 5 + 5 ln 3 Solution: 1. ln a + 1 2 ln b = ln a + ln b 1 2 = ln a + ln b = ln a b 2. Apply log identities 1 and 3, and after log identity 2 to get ln( a + b ) + ln( a - b ) - 2 ln c = ln[( a + b )( a - b )] - ln c 2 = = ln( a 2 - b 2 ) - ln c 2 = ln a 2 - b 2 c 2 3. ln 5 + 5 ln 3 = ln 5 + ln 3 5 = ln(5 · 3 5 ) = ln(1215)
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28 J. S´ anchez-Ortega 2.8 More about Exponential Growth and Decay Models Many natural processes involve quantities that increase or decrease at a rate proportional to their size. For example, the value of an investment bearing
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