Properties of Logarithms

  1. Properties of Logarithms and their proofs

    Property 1:  

    logbxy = ylogbx


    Proof:
      

    We have

            logbxy = logb(blogb(x))y  

            =   logb(bylogb(x)) = ylogbx


    Property 2:  

    logb(xy) =  logbx +  logby


    Property 3:   

    logb(x/y) =  logbx -  logby


    Exercise: 
     

    Prove properties 2 and 3.


  2. Examples

    Expand 

            ln(root(3x^3))

    Solution:  

    We have 

            ln(3x3)1/2 = 1/2 ln(3x3)                (Property 1)

            = 1/2ln3 + 1/2lnx3                       (Property 2)                          

            = 1/2ln3 + 3/2lnx.                        (Property 1)

    Exercises:  Expand the following:

    1. log[(x2(x - 4)5)/100]

    2. log3(sqrt(x5/9))

    Example:    

    Write the following with only one logarithm:

            3log4x - 5log4(x2 + 1) + 2log4x2 

    Solution:   

    We use the properties:    

            log4x3 - log4(x2 + 1)5 + log4(x2)       (Property 1)

            =    log4[x3/(x2 + 1)5] + log4(x4       (Property 3)

            =    log4[x3x4/(x2 + 1)5                    (Property 2)

            =    log4[x7/(x2 + 1)5]                        (A Property of Exponents)

    Exercises:  

    Write the following with only one logarithm:

    1. 2log3x - 2log3sqrt(x) + 5log31/x

    2. logx - 2log(x - 1) + log(x + 1)



  3. Application


    The Rictor scale for earthquakes is as follows:  if I is the intensity of an earthquake and I0 is the intensity of the shaking without an earthquake, then the magnitude R of an earthquake is defined by  

            R = log[I/I0]

    The Loma Prieta quake measured 7.1 on the Rictor scale and the Hokkaido quake measured 8.2.  How many times more intense was the Hokkaido quake?

    Solution

    Let 

            IL = The intensity of the Loma Prieta quake

    and

            IH = The intensity of the Hokkaido quake

            We write

            log(IH/IL)  = log(IH/I0 / IL/I0)

            =  log(IH/I0) - log(IL/I0)

            =  8.2 - 7.1  =  1.1

    By exponentiating both sides with base ten, we get

            IH/IL  = 101.1  =  12.6

    We can conclude that the Hokkaido quake was more than 12 times more intense than the Loma Prieta quake.

 



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