# 1140 L22 - = 1 ; b 6 = 1 and x be positive real numbers....

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Lecture 22: Section 3.3 Properties of Logarithms The Natural Logarithmic Function y = ln x if and only if ln 1 = ln e = e ln x = for x > 0 ln( e x ) = for all real x Properties of Logarithms Let u; v and a be positive real numbers with a 6 = 1 and n be any real number. The following properties hold: 1. log a ( uv ) = 2. log a ± u v ² = 3. log a u n = Proof:

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Caution: log a ( u + v ) 6 = log a u + log a v (log a u ) n 6 = n log a u ex. Evaluate: 1) log 4 2 + log 4 32 2) log 2 80 ± log 2 5 3) ± 1 3 log 4 8 ex. Rewrite and simplify if possible: 1) ln(2 + e x ) 2) log 2 ( x ± y ) 3) log 3 x log 3 y ; y 6 = 1
4) ln ± 1 3 p e ² 5) log 9 4 p 9 3 ! 6) 2 4 log 2 x 7) ln p x 3 e x ± 1 x + 1

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ex. Write as a single logarithm: 1 2 [ln( x ± 5) + ln x ] ± ln(2 y ) Practice 1) ln r x 3 y z 2) log 2 x ± log( x + 1) ± 1 3 log(3 x + 7) Answer: 1) 3 2 ln x + 1 2 ln y ± 1 2 ln z 2) log ± 2 x ( x +1) 3 p 3 x +7 ²
Change of Base Formula Let a 6

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Unformatted text preview: = 1 ; b 6 = 1 and x be positive real numbers. Then log a x = log x log a or log a x = ln x ln a Proof: NOTE: Most calculators have both log and ln keys. By using the Change of Base Formula, we can 1. Evaluate logarithms to other bases. 2. Graph logarithms to other bases. ex. Given ln 3 = 1 : 1 and ln 5 = 1 : 6. Use Change of Base Formula to nd log 3 5 and ln 45. Practice 1. Evaluate log p 3 p 4 + e 2. Solve for x : log 3 x = log 9 (2 x 1) Answer: 1) ln(4+ e ) ln 3 19 11 2) x = 1 only...
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## This note was uploaded on 07/08/2011 for the course MAC 1140 taught by Professor Williamson during the Spring '08 term at University of Florida.

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1140 L22 - = 1 ; b 6 = 1 and x be positive real numbers....

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