Write each expression as a sum and/or difference of logarithms. Express powers as factors: ln ⁡ [ ( x − 4 ) 2 x 2 − 1 ] 2 / 3 the equation inside the brackets is suppose to be to the 2/3 and they also give x&gt;4 with the equation.

Question

Write each expression as a sum and/or difference of logarithms. Express powers as factors:  ln  ⁡  [            (      x      −      4              )                  2                                    x                  2                    −      1            ]          2              /            3      the equation inside the brackets is suppose to be to the 2/3 and they also give x&amp;gt;4 with the equation.

Jacey Bowers · Accepted Answer

First a review of log properties.1.Powers inside a log come down and can be expressed as a factor in front of the log. For example,   ln  ⁡  (      x    2    )  =  2  ln  ⁡  (  x  )     a  n  d     ln  ⁡  (      x    )  =      1    2    ln  ⁡  (  x  ).2. Multiplication/Division inside a logcan be expressed as the sum or difference of two logs, respectively. For example,   ln  ⁡  (  2  x  )  =  ln  ⁡  2  +  ln  ⁡  x  ,     a  n  d     ln  ⁡  (  x      /    2  )  =  ln  ⁡  x  −  ln  ⁡  2.With these two rules in mind, we proceed:  ln  ⁡  [            (      x      −      4              )                  2                                    x                  2                    −      1            ]          2              /            3      Bring down the 2/3:      2    3    [  ln  ⁡  (            (      x      −      4              )                  2                                    x                  2                    −      1        )  ]Separate the division into two logs:      2    3    [  ln  ⁡  (  x  −  4      )          2        −  ln  ⁡  (      x          2        −  1  )  ]Bring the 2 down on the first log. Also, we can factor       x          2        −  1 into (x+1)(x-1):      2    3    [  2  ln  ⁡  (  x  −  4  )  −  ln  ⁡  [  (  x  +  1  )  (  x  −  1  )  ]  ]Split the multiplication into the sum of two logs. But be careful! That minus sign on the original log will end up applying to BOTH ofthese new logs!:      2    3    [  2  ln  ⁡  (  x  −  4  )  −  [  ln  ⁡  (  x  +  1  )  +  ln  ⁡  (  x  −  1  )  ]  ]Distribute the negative sign:      2    3    [  2  ln  ⁡  (  x  −  4  )  −  ln  ⁡  (  x  +  1  )  −  ln  ⁡  (  x  −  1  )  ]

Macy Villanueva · Accepted Answer

we need to apply some formula to solve this:-1)   ln  ⁡  (  A  )  B  =  B  ∗  ln  ⁡  (  A  )2)   ln  ⁡  (  A      /    B  )  =  ln  ⁡  (  A  )  −  ln  ⁡  (  B  )3)   ln  ⁡  (  A  ∗  B  )  =  ln  ⁡  (  A  )  +  ln  ⁡  (  B  )now given,   ln  ⁡  {  [  (  x  −  4      )    2        /    (      x    2    −  1  )      }          2              /            3        =  2      /    3  ∗  [  ln  ⁡  {  (  x  −  4      )    2        /    (      x    2    −  1  )  }  ]...............applying (1)  =  2      /    3  ∗  [  ln  ⁡  {  (  x  −  4      )    2    }  −  ln  ⁡  {  (      x    2    −  1  )  }  ]........applying (2)  =  2      /    3  ∗  [  2  ∗  ln  ⁡  (  x  −  4  )  −  ln  ⁡  {  (  x  −  1  )  ∗  (  x  +  1  )  }  ] ..applying(1)  =  2      /    3  ∗  [  2  ∗  ln  ⁡  (  x  −  4  )  −  ln  ⁡  (  x  −  1  )  +  ln  ⁡  (  x  +  1  )  ] ..applying (3)Answer:-   2      /    3  ∗  [  2  ∗  ln  ⁡  (  x  −  4  )  −  ln  ⁡  (  x  −  1  )  +  ln  ⁡  (  x  +  1  )  ]

Write each expression as a sum and/or difference of logarithms. Express powers as factors: ln[((x-4)^2)/(x^2 -1)]^(2/3) the equation inside the brackets is suppose to be to the 2/3 and they also give x>4 with the equation.

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