Properties of Logarithms

Section 9.0B

 

Find x for        

        log332 = x       

rewrite            

        3x = 32  

therefore 

        x = 2


        Log525 = x                 

        5x =25             

        x = 2


        Ln e3 = x         

        ex = e3             

        x = 3


        Log 10 = x      

        10x = 10          

        x = 1


 

Inverse properties:

        log10x = x

        ln ex = x

 

Do not confuse log with ln.  

        log ex

and 

        ln 10x x ,

 

More inverses (the functions undo each other).   

For example 

        ( )2 = x

          elnx = x

          10logx = x

 

Example 1:         Solving exponential equations.

 

a)         10x = 3                                                Take the common log of both sides

Log10x = log 3                                   Inverse property

X = log 3 = 0.47712155               Use calculator

 

b)         2e3x = 5          the base is e, but before taking ln of both sides, isolate e3x

            e3x  = 5/2                    take ln of both sides

            ln e3x = ln (5/2)           Inverse property

            3x = ln (5/2)                Solve for x

            x = ln(5/2) = 0.305430244
                     3

 

Property:  Exponent becomes multiplier

        log Ar = r log A                       

        ln Ar = r ln A

 

Proof:  

Let                               

        log A = y

rewrite                        

        10y      = A

raise both sides to n             

        (10y)n = An

take log of both sides           

        log 10ny = log An

Inverse Property                   

        ny = log An                  

but 

        y = log A               (given)

Therefore                               

        n(y) = n(log A)


Example 3:              

Solve  

        32x = 5                         

The base is neither 10 nor e, but we can still take common log of both sides and use multiplier rule.

            Log 32x = log 5

            2xlog 3 = log 5

             x  = log5/2log3 = 0.73248676

 

Division becomes subtraction property.

  log A/B = log A log B

  ln A/B = ln A ln B

 

Note:     log A/log B log A log B

 

Proof: 

  Let                                        a = log A         b = log B

Rewrite                                  10a = A           10b = B

Divide                                     A/B = 10a/ 10b

Property of exponents             A/B = 10a b

Take log of both                      log A/B = log 10a b

Inverse Property                      log A/B = a b

Given                                       log A/B = log A log B

 

EX 5:  

Rewrite           

a)          log 5x/3                 

without fraction

          log 5x log 3

 

b)        log 3 log 2x        

as a single log

         log 3/2x

 

Multiplication becomes addition property.

 

        log (AB) = log A + log B                     ln (AB) = ln A + ln B

 

Extra Credit: Prove this property.

 

Exercise 6:  

Rewrite           

a)  log 4x                    

as separate logs

        Log 4 + log x

 

b)  log 5 + log 3x        

as a single log

        log 15x

 

Exercise 7: 

Using all three properties (multiplier, division, addition) solve the following.

 

Log x + log 5 = 3                               Addition property

Log 5x = 3                                          Rewrite into exponential form

103 = 5x                                              Simplify

1000 = 5x                                           Solve for x

1000/5 = x = 200


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