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10 Pages

### L21

Course: MATH MAC1147, Summer 2008
School: University of Florida
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Word Count: 285

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21: Lecture Section 3.2 Logarithmic Functions Recall the graph of exponential function ( ) = &gt; 1: , (0,1) Since ( ) is one-to-one, it has an inverse function. Def. The logarithmic function with base , where &gt; 0 and = 1 is written ( ) = log and is dened by the relationship = log if and only if ex. Write in exponential form: 1) log3 1 9 = 2 2) log (3 + 1) = 2 ex. Write in logarithmic form:...

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21: Lecture Section 3.2 Logarithmic Functions Recall the graph of exponential function ( ) = > 1: , (0,1) Since ( ) is one-to-one, it has an inverse function. Def. The logarithmic function with base , where > 0 and = 1 is written ( ) = log and is dened by the relationship = log if and only if ex. Write in exponential form: 1) log3 1 9 = 2 2) log (3 + 1) = 2 ex. Write in logarithmic form: 41.5 = 8 ex. Evaluate: 1) log2 64 = 2) log3 1 = 3) log3 3 = 4) log10 1 1000 5) log16 4 = 6) log2(1) = 7) log2 0 = = Properties of Logarithms 1. Recall: If That is, = log , then = > 0. 2. log 1 = 3. log = 4. Inverse Properties: log log = = for all real number for >0 5. One-to-One Properties: If log = log , then ex. Evaluate: 1) 2log2 3 = 2) log5 1 = 5 ex. Solve: log2( 3) = log2 9 Graphs of Logarithmic Functions ex. Sketch = 2 and ex. Sketch = 1 2 = log2 . and = log 1 2 Properties of the graph of ( ) = log Compare ( ) = log and 1 ( ) = log 1. Domain: 2. Range: 3. Intercept: 4. Asymptote: 5. increasing if decreasing if 6. points on the graph ex. Graph ( ) = log3( + 1) ( )= The )= : 1 ( Natural Logarithmic Function = log = ln if and only if Note the following: ln 1 = ln = Inverse Properties: ln = ln( ) = One-to-One Property: If ln = ln , then ex. Evaluate: 1) ln(2 +3) 2) ln 1 = = ex. Solve: ln( 2 ) = ln 6 ex. Graph and nd the domain and vertical asymptote of ( ): 1) ( ) = ln 2) ( ) = ln( 2) + 1 3) ( ) = ln( ) + 2 Common Logarithm Function = log10 = log if and only if ex. Evaluate: log 1 = log 10 = log 10000 = 1 log = 10 Applications ex. The loudness level of a sound, given by = 10 log 1012 , in decibels, is , where is the intensity of a sound in watts per square meter. 1) Determine the decibel level of a sound with an intensity of 102 watt per square meter. 2) Determine the decibel level of a sound with an intensity of 1 watts per square meter. 3) The intensity of a sound in part (2) is 100 times as great as the intensity in part (1). By how much is the decible level increased? Practice. Determine the domain of each function. 1) ( ) = log( 2 2) ( ) = log 3 +3 2 ) 3) ( ) = ln 2 Answer. 1) (, 0) (2, ) 2) (3, 3) 3) (, 2) (2, )
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University of Florida - MATH - MAC1147
Lecture 23: Section 3.4Exponential and Logarithmic EquationsTwo basic strategies for solving exponential orlogarithmic equations:1. One-to-One Properties=logif and only if= logif and only if2. Inverse Propertieslog=log=ex. Solve for :1) 2
University of Florida - MATH - MAC1147
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University of Florida - MATH - MAC1147
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University of Florida - MATH - MAC1147
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University of Florida - MATH - MAC1147
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University of Florida - MATH - MAC1147
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University of Florida - MATH - MAC1147
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UCSB - ECON - 171
University of Florida - MAC - 2233
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University of Florida - MAC - 2233
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UCSB - ECON - 171
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University of Florida - MAC - 2233
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University of Florida - MAC - 2233
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University of Florida - MAC - 2233
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University of Florida - MAC - 2233
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University of Florida - MAC - 2233
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UCSB - ECON - 171
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University of Florida - MAC - 2233
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UCSB - ECON - 171
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University of Florida - MAC - 2233
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University of Florida - MAC - 2233
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