Problem: Log function with common base equationln(x-2)-ln(x+9)=ln(x-1)-ln(x+14)I dropped the ln's and tried solving by using the quotient property but I think im doing the fractions wrong.can someone please explain how to get the answer? (solving for x)Thank you

Question

Problem: Log function with common base equationln(x-2)-ln(x+9)=ln(x-1)-ln(x+14)I dropped the ln&#039;s and tried solving by using the quotient property but I think im doing the fractions wrong.can someone please explain how to get the answer? (solving for x)Thank you

hrikalegt15 · Accepted Answer

ln  ⁡  (  x  −  2  )  −  ln  ⁡  (  x  +  9  )  =  ln  ⁡  (  x  −  1  )  −  l  n  (  x  +  14  )  ⇒  ln  ⁡      (                  x        −        2                    x        +        9              )    =  ln  ⁡      (                  x        −        1                    x        +        14              )  , but   ln  ⁡  (  x  ) is bijective and so            x      −      2              x      +      9        =            x      −      1              x      +      14      and solving that we have:  (  x  −  2  )  (  x  +  14  )  =  (  x  −  1  )  (  x  +  9  )  ⇒  x  =  19      /    4and testing that solution at the original equation we see that it holds.

ureq8 · Accepted Answer

I&#039;m approaching this within the context of the natural log, until we have a clear equation with which to solve for   x  ln  ⁡  (  x  −  2  )  −  ln  ⁡  (  x  +  9  )  =  ln  ⁡  (  x  −  1  )  −  ln  ⁡  (  x  +  14  )    ⟺    ln  ⁡      (                  x        −        2                    x        +        9              )    =  ln  ⁡      (                  x        −        1                    x        +        14              )      ⟺    ln  ⁡      (                  x        −        2                    x        +        9              )    −  ln  ⁡      (                  x        −        1                    x        +        14              )    =  0    ⟺    ln  ⁡      (                            x          −          2                          x          +          9                                      x          −          1                          x          +          14                      )    =  0    ⟺    ln  ⁡      (                  (        x        −        2        )        (        x        +        14        )                    (        x        −        1        )        (        x        +        9        )              )    =  0Noting that   ln  ⁡  (  f  (  x  )  )  =  0    ⟺    f  (  x  )  =  1, we have   f  (  x  )  =            g      (      x      )              h      (      x      )        =  1, meaning we must have                    (solve for x)                    (        x        −        2        )        (        x        +        14        )        =        (        x        −        1        )        (        x        +        9        )            to arrive at   ln  ⁡  (  1  )  =  0

Problem: Log function with common base equation ln(x-2)-ln(x+9)=ln(x-1)-ln(x+14) I dropped the ln's and tried solving by using the quotient property but I think im doing the fractions wrong. can someone please explain how to get the answer? (solving for x) Thank you

Open question

Answer & Explanation

New Questions in Pre-Algebra