Properties of Logarithms

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.

 



Back to the College Algebra Part II (Math 103B) Site

Back to the LTCC Math Department Page

e-mail Questions and Suggestions