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 
                    
                    
                    
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                   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. 
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                   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. 
                    
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                   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
                        
                        
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