Properties of Logarithms

1. Properties of Logarithms and their proofs
   
   

   Property 1:   logbxy = ylogbx

   
   Proof:   
   
   We have

           log_bx^y = log_b(b^logb(x))^y  
   
           =   log_b(b^ylogb(x)) = ylog_bx
   
   
   

   Property 2:   logb(xy) =  logbx +  logby

   
   

   Property 3:    logb(x/y) =  logbx -  logby

   
   Exercise:   
   
   Prove properties 2 and 3.

   
   

2. Examples

   Expand 
   
          

   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)²        (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 
   
           I_L = The intensity of the Loma Prieta quake
   
   and
   
           I_H = The intensity of the Hokkaido quake
   
           We write
   
           log(I_H/I_L)  = log(I_H/I₀ / I_L/I₀)
   
           =  log(I_H/I₀) - log(I_L/I₀)

           =  8.2 - 7.1  =  1.1

   By exponentiating both sides with base ten, we get
   
           I_H/I_L  = 10^1.1  =  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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