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File: Lecture 19
math 110 lecture 19 ch 4 3 4 4 part i logarithmic function logarithmic equations change of base formula for any logarithmic bases a and b and any positive number ...

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                   Math 110                                                                                          Lecture #19         
                   CH. 4.3-4.4 (PART I). Logarithmic Function. Logarithmic equations.                                       
                    
                   ƒ  Change-of-Base Formula. 
                    
                    
                                    For any logarithmic bases a and b, and any 
                                    positive number M, 
                                     
                                                      log        M=logaM 
                                                              b            log b
                                                                                 a
                    
                    
                   Problem #1. 
                    
                   Use your calculator to find the following logarithms. 
                   Show your work with Change-of-Base Formula. 
                    
                        log 10                             log 9                              log 11
                   a)                  b)     c)   
                             2                                   1                                  7
                                                                 3
                    
                    
                   ƒ  Using the Change-of-Base Formula, we can graph 
                       Logarithmic Functions with an arbitrary base.  
                       Example:  
                    
                   log x = ln x
                         2        ln2
                                  logx 
                   log2 x = log2
                    
                                                                                                y = log2 x 
                    
                    
                    
                                                                                                                                  1
                   Math 110                                                                                          Lecture #19         
                   CH. 4.3-4.4 (PART I). Logarithmic Function. Logarithmic equations.                                       
                   ƒ  Properties of Logarithms.  
                         
                    
                   If b, M, and N are positive real numbers, b ≠1, p, x are real 
                   numbers, then 
                    
                      1. log MNM=+log                          log N
                                                                               product rule 
                                 bbb
                    
                    
                      2. log M =−log M                       log N     quotient rule 
                                 bbb
                                    N
                    
                    
                           log Mp = plog M
                      3.                                                      power rule 
                                 bb
                    
                    
                           ⎧            x
                      4. ⎪logbbx=                                       inverse property of logarithms 
                           ⎨ log x
                                   b
                             bx=>,0x
                           ⎪
                           ⎩
                    
                    
                      5. log M ==log N if and only if M                                      N
                                 bb . 
                           This property is the base for solving Logarithmic  
                           Equations in form  log gx=log hx. 
                                                                          ( )                  ( )
                                                                    bb
                    
                   Properties 1-3 may be used for Expanding and Condensing 
                   Logarithmic expressions. 
                    
                    
                    
                    
                    
                   ƒ  Expanding and Condensing Logarithmic expressions. 
                                                                                                                                  2
                   Math 110                                                                                          Lecture #19         
                   CH. 4.3-4.4 (PART I). Logarithmic Function. Logarithmic equations.                                       
                    
                   Problem #2. 
                    
                   Express each of the following expressions as a single 
                   logarithm  
                   whose coefficient is equal to 1.  
                        
                       a) 1⎡⎤
                                               ++ −−  
                                 3log xx1                2log             3       log7
                                          ()()
                            5⎣⎦
                    
                            11
                               ⎡⎤
                                          ++ −+
                       b)        ln xx1              2ln           1            ln x 
                                     ()()
                               ⎣⎦
                            23
                    
                            11
                                                       ⎡⎤
                       c)               +− + + 
                               ln xx3                    ln        3ln x1
                                   () ()
                                                       ⎣⎦
                            25
                    
                            1⎡⎤
                                            −+ +−  
                       d)        log xx2                2log             2       log5
                                        ()()
                            2⎣⎦
                    
                   Problem #3. 
                    
                   Expand a much as possible each of the following. 
                    
                                   x2y
                       a) log z5  
                    
                    
                                    x3y
                       b) ln 4               
                                     z3
                    
                    
                   ƒ  Solving Logarithmic Equations. 
                    
                                                                                                                                  3
                   Math 110                                                                                          Lecture #19         
                   CH. 4.3-4.4 (PART I). Logarithmic Function. Logarithmic equations.                                       
                       1. Solving the Simplest Logarithmic Equation (SLE). 
                            Given: log xa=                                           a
                                               b           , b > 0, b ≠1,   is any real number. 
                            According the definition of the logarithm this equation is 
                            equivalent to  x = ba. 
                    
                        2. According to properties of logarithms, if   
                           log M = log N , then M = N. 
                                  bb
                    
                            Remember, check is part of solution for 
                            Logarithmic Equations. 
                    
                    
                   Problem #4. Solve the following Logarithmic Equations. 
                    
                       a) log2 x =5 
                        
                        
                        
                       b)                                
                            log        x−=25
                                   3 ()
                        
                        
                        
                       c)               2                      
                             log xx−=log6
                                   ()
                        
                        
                        
                        
                        
                        
                        
                        
                       d)                                    
                             log        x+=4           −3
                                    1 ()
                                    2
                        
                        
                                                                                                                                  4
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...Math lecture ch part i logarithmic function equations change of base formula for any bases a and b positive number m log logam problem use your calculator to find the following logarithms show work with c using we can graph functions an arbitrary example x ln logx y properties if n are real numbers p then mnm product rule bbb quotient mp plog power bb logbbx inverse property bx only this is solving in form gx hx may be used expanding condensing expressions express each as single logarithm whose coefficient equal xx d expand much possible xy z simplest equation sle given xa according definition equivalent ba remember check solution solve...

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