Simplify $frac56logleft(frac54right) - frac16log(2)$ to $logleft(frac54right) - frac16logleft(frac52right)$

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I'm trying to bring this expression:



$$frac56logleft(frac54right) - frac16log(2)$$



To this one:



$$logleft(frac54right) - frac16logleft(frac52right)$$



Where $log$ is the natural algorithm.



I know the two expressions are equal (checked with wolfram) but I really can't find the correct passages... Could you please help me?










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    up vote
    3
    down vote

    favorite












    I'm trying to bring this expression:



    $$frac56logleft(frac54right) - frac16log(2)$$



    To this one:



    $$logleft(frac54right) - frac16logleft(frac52right)$$



    Where $log$ is the natural algorithm.



    I know the two expressions are equal (checked with wolfram) but I really can't find the correct passages... Could you please help me?










    share|cite|improve this question

























      up vote
      3
      down vote

      favorite









      up vote
      3
      down vote

      favorite











      I'm trying to bring this expression:



      $$frac56logleft(frac54right) - frac16log(2)$$



      To this one:



      $$logleft(frac54right) - frac16logleft(frac52right)$$



      Where $log$ is the natural algorithm.



      I know the two expressions are equal (checked with wolfram) but I really can't find the correct passages... Could you please help me?










      share|cite|improve this question















      I'm trying to bring this expression:



      $$frac56logleft(frac54right) - frac16log(2)$$



      To this one:



      $$logleft(frac54right) - frac16logleft(frac52right)$$



      Where $log$ is the natural algorithm.



      I know the two expressions are equal (checked with wolfram) but I really can't find the correct passages... Could you please help me?







      algebra-precalculus logarithms






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      edited Sep 22 at 10:53









      user21820

      36.7k441143




      36.7k441143










      asked Sep 21 at 23:28









      Alessio Martorana

      1057




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          5 Answers
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          up vote
          6
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          accepted










          $frac56logleft(frac54right)-frac16log(2)=logleft(frac54right)-frac16logleft(frac54right)-frac16log(2)$



          $= logleft(frac54right)-(frac16logleft(frac54right)+frac16log(2))$



          $=logleft(frac54right)-frac16logleft(frac5cdot24right)$



          $=logleft(frac54right)-frac16logleft(frac52right)$



          I should add: Subtracting logs is equivalent to division. The reason for potential problems here is $fracfracabc = fracabcdot c$ and not $fraca(fracbc)$






          share|cite|improve this answer





























            up vote
            6
            down vote













            Hint: Begin with$$frac56logleft(frac54right)-frac16log(2)=logleft(frac54right)-frac16logleft(frac54right)-frac16log(2).$$






            share|cite|improve this answer




















            • Ty very much for the hint, I was on the right road, but missed the last passage that, imho, Phil stated very simply and clear
              – Alessio Martorana
              Sep 22 at 0:25

















            up vote
            3
            down vote













            We have



            $$frac56logleft(frac54right) - frac16log 2=overbracefrac56logleft(frac54right)+colorredfrac16logleft(frac54right) -overbracecolorredfrac16logleft(frac54right)- frac16log 2=$$



            $$=logleft(frac54right)-frac16 logleft(frac52right)$$



            indeed recall that $log A+ log B= log AB$ and therefore



            $$-frac16logleft(frac54right)- frac16log 2=-frac16left[logleft(frac54right)+log 2right]=-frac16logleft(frac54cdot 2right)=-frac16logleft(frac52right)$$






            share|cite|improve this answer





























              up vote
              2
              down vote













              Alt. hint:   by brute force, using only $,log fracab = log a - log b,$ and $,log a^n = n log a,$:



              $$small
              frac56logfrac54 - frac16log 2 = frac16left(5 log 5 - 5 log 4-log 2right) = frac16left(5 log 5 - 10 log 2-log 2right) = frac16left(5 log 5 - 11 log 2right)
              $$



              Now do the same for $,log frac54 - frac16log frac52,$ and compare.






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                $$fraccolorblue56log frac54 - frac16log 2 = fraccolorblue6-16logfrac54 - frac16log2 = log frac54 - frac16left(colorgreenlogfrac54 + log 2 right)$$$$ = log frac54 - frac16colorgreen log left( frac54 cdot 2 right) = log frac54 - frac16 log frac52 $$






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                  5 Answers
                  5






                  active

                  oldest

                  votes








                  5 Answers
                  5






                  active

                  oldest

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                  active

                  oldest

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                  up vote
                  6
                  down vote



                  accepted










                  $frac56logleft(frac54right)-frac16log(2)=logleft(frac54right)-frac16logleft(frac54right)-frac16log(2)$



                  $= logleft(frac54right)-(frac16logleft(frac54right)+frac16log(2))$



                  $=logleft(frac54right)-frac16logleft(frac5cdot24right)$



                  $=logleft(frac54right)-frac16logleft(frac52right)$



                  I should add: Subtracting logs is equivalent to division. The reason for potential problems here is $fracfracabc = fracabcdot c$ and not $fraca(fracbc)$






                  share|cite|improve this answer


























                    up vote
                    6
                    down vote



                    accepted










                    $frac56logleft(frac54right)-frac16log(2)=logleft(frac54right)-frac16logleft(frac54right)-frac16log(2)$



                    $= logleft(frac54right)-(frac16logleft(frac54right)+frac16log(2))$



                    $=logleft(frac54right)-frac16logleft(frac5cdot24right)$



                    $=logleft(frac54right)-frac16logleft(frac52right)$



                    I should add: Subtracting logs is equivalent to division. The reason for potential problems here is $fracfracabc = fracabcdot c$ and not $fraca(fracbc)$






                    share|cite|improve this answer
























                      up vote
                      6
                      down vote



                      accepted







                      up vote
                      6
                      down vote



                      accepted






                      $frac56logleft(frac54right)-frac16log(2)=logleft(frac54right)-frac16logleft(frac54right)-frac16log(2)$



                      $= logleft(frac54right)-(frac16logleft(frac54right)+frac16log(2))$



                      $=logleft(frac54right)-frac16logleft(frac5cdot24right)$



                      $=logleft(frac54right)-frac16logleft(frac52right)$



                      I should add: Subtracting logs is equivalent to division. The reason for potential problems here is $fracfracabc = fracabcdot c$ and not $fraca(fracbc)$






                      share|cite|improve this answer














                      $frac56logleft(frac54right)-frac16log(2)=logleft(frac54right)-frac16logleft(frac54right)-frac16log(2)$



                      $= logleft(frac54right)-(frac16logleft(frac54right)+frac16log(2))$



                      $=logleft(frac54right)-frac16logleft(frac5cdot24right)$



                      $=logleft(frac54right)-frac16logleft(frac52right)$



                      I should add: Subtracting logs is equivalent to division. The reason for potential problems here is $fracfracabc = fracabcdot c$ and not $fraca(fracbc)$







                      share|cite|improve this answer














                      share|cite|improve this answer



                      share|cite|improve this answer








                      edited Sep 22 at 1:45

























                      answered Sep 22 at 0:15









                      Phil H

                      2,7412311




                      2,7412311




















                          up vote
                          6
                          down vote













                          Hint: Begin with$$frac56logleft(frac54right)-frac16log(2)=logleft(frac54right)-frac16logleft(frac54right)-frac16log(2).$$






                          share|cite|improve this answer




















                          • Ty very much for the hint, I was on the right road, but missed the last passage that, imho, Phil stated very simply and clear
                            – Alessio Martorana
                            Sep 22 at 0:25














                          up vote
                          6
                          down vote













                          Hint: Begin with$$frac56logleft(frac54right)-frac16log(2)=logleft(frac54right)-frac16logleft(frac54right)-frac16log(2).$$






                          share|cite|improve this answer




















                          • Ty very much for the hint, I was on the right road, but missed the last passage that, imho, Phil stated very simply and clear
                            – Alessio Martorana
                            Sep 22 at 0:25












                          up vote
                          6
                          down vote










                          up vote
                          6
                          down vote









                          Hint: Begin with$$frac56logleft(frac54right)-frac16log(2)=logleft(frac54right)-frac16logleft(frac54right)-frac16log(2).$$






                          share|cite|improve this answer












                          Hint: Begin with$$frac56logleft(frac54right)-frac16log(2)=logleft(frac54right)-frac16logleft(frac54right)-frac16log(2).$$







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Sep 21 at 23:31









                          José Carlos Santos

                          126k17102189




                          126k17102189











                          • Ty very much for the hint, I was on the right road, but missed the last passage that, imho, Phil stated very simply and clear
                            – Alessio Martorana
                            Sep 22 at 0:25
















                          • Ty very much for the hint, I was on the right road, but missed the last passage that, imho, Phil stated very simply and clear
                            – Alessio Martorana
                            Sep 22 at 0:25















                          Ty very much for the hint, I was on the right road, but missed the last passage that, imho, Phil stated very simply and clear
                          – Alessio Martorana
                          Sep 22 at 0:25




                          Ty very much for the hint, I was on the right road, but missed the last passage that, imho, Phil stated very simply and clear
                          – Alessio Martorana
                          Sep 22 at 0:25










                          up vote
                          3
                          down vote













                          We have



                          $$frac56logleft(frac54right) - frac16log 2=overbracefrac56logleft(frac54right)+colorredfrac16logleft(frac54right) -overbracecolorredfrac16logleft(frac54right)- frac16log 2=$$



                          $$=logleft(frac54right)-frac16 logleft(frac52right)$$



                          indeed recall that $log A+ log B= log AB$ and therefore



                          $$-frac16logleft(frac54right)- frac16log 2=-frac16left[logleft(frac54right)+log 2right]=-frac16logleft(frac54cdot 2right)=-frac16logleft(frac52right)$$






                          share|cite|improve this answer


























                            up vote
                            3
                            down vote













                            We have



                            $$frac56logleft(frac54right) - frac16log 2=overbracefrac56logleft(frac54right)+colorredfrac16logleft(frac54right) -overbracecolorredfrac16logleft(frac54right)- frac16log 2=$$



                            $$=logleft(frac54right)-frac16 logleft(frac52right)$$



                            indeed recall that $log A+ log B= log AB$ and therefore



                            $$-frac16logleft(frac54right)- frac16log 2=-frac16left[logleft(frac54right)+log 2right]=-frac16logleft(frac54cdot 2right)=-frac16logleft(frac52right)$$






                            share|cite|improve this answer
























                              up vote
                              3
                              down vote










                              up vote
                              3
                              down vote









                              We have



                              $$frac56logleft(frac54right) - frac16log 2=overbracefrac56logleft(frac54right)+colorredfrac16logleft(frac54right) -overbracecolorredfrac16logleft(frac54right)- frac16log 2=$$



                              $$=logleft(frac54right)-frac16 logleft(frac52right)$$



                              indeed recall that $log A+ log B= log AB$ and therefore



                              $$-frac16logleft(frac54right)- frac16log 2=-frac16left[logleft(frac54right)+log 2right]=-frac16logleft(frac54cdot 2right)=-frac16logleft(frac52right)$$






                              share|cite|improve this answer














                              We have



                              $$frac56logleft(frac54right) - frac16log 2=overbracefrac56logleft(frac54right)+colorredfrac16logleft(frac54right) -overbracecolorredfrac16logleft(frac54right)- frac16log 2=$$



                              $$=logleft(frac54right)-frac16 logleft(frac52right)$$



                              indeed recall that $log A+ log B= log AB$ and therefore



                              $$-frac16logleft(frac54right)- frac16log 2=-frac16left[logleft(frac54right)+log 2right]=-frac16logleft(frac54cdot 2right)=-frac16logleft(frac52right)$$







                              share|cite|improve this answer














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                              share|cite|improve this answer








                              edited Sep 22 at 6:00

























                              answered Sep 22 at 0:16









                              gimusi

                              1




                              1




















                                  up vote
                                  2
                                  down vote













                                  Alt. hint:   by brute force, using only $,log fracab = log a - log b,$ and $,log a^n = n log a,$:



                                  $$small
                                  frac56logfrac54 - frac16log 2 = frac16left(5 log 5 - 5 log 4-log 2right) = frac16left(5 log 5 - 10 log 2-log 2right) = frac16left(5 log 5 - 11 log 2right)
                                  $$



                                  Now do the same for $,log frac54 - frac16log frac52,$ and compare.






                                  share|cite|improve this answer
























                                    up vote
                                    2
                                    down vote













                                    Alt. hint:   by brute force, using only $,log fracab = log a - log b,$ and $,log a^n = n log a,$:



                                    $$small
                                    frac56logfrac54 - frac16log 2 = frac16left(5 log 5 - 5 log 4-log 2right) = frac16left(5 log 5 - 10 log 2-log 2right) = frac16left(5 log 5 - 11 log 2right)
                                    $$



                                    Now do the same for $,log frac54 - frac16log frac52,$ and compare.






                                    share|cite|improve this answer






















                                      up vote
                                      2
                                      down vote










                                      up vote
                                      2
                                      down vote









                                      Alt. hint:   by brute force, using only $,log fracab = log a - log b,$ and $,log a^n = n log a,$:



                                      $$small
                                      frac56logfrac54 - frac16log 2 = frac16left(5 log 5 - 5 log 4-log 2right) = frac16left(5 log 5 - 10 log 2-log 2right) = frac16left(5 log 5 - 11 log 2right)
                                      $$



                                      Now do the same for $,log frac54 - frac16log frac52,$ and compare.






                                      share|cite|improve this answer












                                      Alt. hint:   by brute force, using only $,log fracab = log a - log b,$ and $,log a^n = n log a,$:



                                      $$small
                                      frac56logfrac54 - frac16log 2 = frac16left(5 log 5 - 5 log 4-log 2right) = frac16left(5 log 5 - 10 log 2-log 2right) = frac16left(5 log 5 - 11 log 2right)
                                      $$



                                      Now do the same for $,log frac54 - frac16log frac52,$ and compare.







                                      share|cite|improve this answer












                                      share|cite|improve this answer



                                      share|cite|improve this answer










                                      answered Sep 22 at 0:19









                                      dxiv

                                      57.3k64799




                                      57.3k64799




















                                          up vote
                                          0
                                          down vote













                                          $$fraccolorblue56log frac54 - frac16log 2 = fraccolorblue6-16logfrac54 - frac16log2 = log frac54 - frac16left(colorgreenlogfrac54 + log 2 right)$$$$ = log frac54 - frac16colorgreen log left( frac54 cdot 2 right) = log frac54 - frac16 log frac52 $$






                                          share|cite|improve this answer
























                                            up vote
                                            0
                                            down vote













                                            $$fraccolorblue56log frac54 - frac16log 2 = fraccolorblue6-16logfrac54 - frac16log2 = log frac54 - frac16left(colorgreenlogfrac54 + log 2 right)$$$$ = log frac54 - frac16colorgreen log left( frac54 cdot 2 right) = log frac54 - frac16 log frac52 $$






                                            share|cite|improve this answer






















                                              up vote
                                              0
                                              down vote










                                              up vote
                                              0
                                              down vote









                                              $$fraccolorblue56log frac54 - frac16log 2 = fraccolorblue6-16logfrac54 - frac16log2 = log frac54 - frac16left(colorgreenlogfrac54 + log 2 right)$$$$ = log frac54 - frac16colorgreen log left( frac54 cdot 2 right) = log frac54 - frac16 log frac52 $$






                                              share|cite|improve this answer












                                              $$fraccolorblue56log frac54 - frac16log 2 = fraccolorblue6-16logfrac54 - frac16log2 = log frac54 - frac16left(colorgreenlogfrac54 + log 2 right)$$$$ = log frac54 - frac16colorgreen log left( frac54 cdot 2 right) = log frac54 - frac16 log frac52 $$







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                                              answered Sep 22 at 0:09









                                              trancelocation

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