Properties of Logarithms

Section 9.0B

Find x for        

        log₃3² = x       

rewrite            

        3^x = 3²  

therefore 

        x = 2

        Log₅25 = x                 

        5^x =25             

        x = 2

        Ln e³ = x         

        e^x = e³             

        x = 3

        Log 10 = x      

        10^x = 10          

        x = 1

Inverse properties:

        log10^x = x

        ln e^x = x

Do not confuse log with ln.  

        log e^x x 

and 

        ln 10^x x ,

More inverses (the functions undo each other).   

For example 

        ( )² = x

          e^lnx = x

          10^logx = x

Example 1:         Solving exponential equations.

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

Log10^x = log 3                                   Inverse property

X = log 3 = 0.47712155…               Use calculator

b)         2e^3x = 5          the base is e, but before taking ln of both sides, isolate e^3x

            e^3x  = 5/2                    take ln of both sides

            ln e^3x = ln (5/2)           Inverse property

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

            x = ln(5/2) = 0.305430244
                     3

Property:  Exponent becomes multiplier

        log A^r = r log A                       

        ln A^r = r ln A

Proof:  

Let                               

        log A = y

rewrite                        

        10^y      = A

raise both sides to n             

        (10^y)ⁿ = Aⁿ

take log of both sides           

        log 10^ny = log Aⁿ

Inverse Property                   

        ny = log Aⁿ                  

but 

        y = log A               (given)

Therefore                               

        n(y) = n(log A)

Example 3:              

Solve  

        3^2x = 5                         

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

            Log 3^2x = 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                                  10^a = A           10^b = B

Divide                                     A/B = 10^a/ 10^b

Property of exponents             A/B = 10^{a – b}

Take log of both                      log A/B = log 10^{a – 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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