if log a +logb+logc/log6= 6..and b-a is a perfect cube and a,b,c are in geometric sequence then a+b+c =??????????

• since a,b,c are in geometric sequence so,
  if a=m then,
  b=mn and
  c= mn^2
  therefor,
  m*mn*mn^2= m^3*n^3 = (mn)^3
  now acording to question,
  log a+log b+log c/log 6=6
  so,
  logabc/log6=6
  logabc 6=6
  6^6=abc
  now
  put the value of abc ,
  6^6= (mn)^3
  mn= 6*6=36=b
  now acccording to question b-a=Perfect cube
  so a=9=m
  therefor n=4
  therefor c=144
  so a+b+c=9+36+144=189 ans.
• 10 years agoHelpfull: Yes(51) No(2)
• loga+logb+logc/log6=6
  so,logabc/log6=6
  abc=6^6
  as,
  b^2=ac
  therefore,b^3=6^6
  hence on solving
  b=36
  and b-a is a perfect cube
  and
  36ac=46656
  ac=1296
  and according to condition a=9 as it is perfectly divide the term 1296
  so then c=144
  therefore , 9+36+144=189
  
  
• 9 years agoHelpfull: Yes(9) No(1)
• loga+logb+logc/log6=6
  log(abc)/log6=6
  log(abc)=6log6
  log(abc)=log6^6
  abc=6^6
  a*ar*ar^2=6^6
  (ar)^3=6^6
  ar=6^2=36....also given b-a=ar-a=36-a=8(as perfect sq.)
  a=28
  a+b+c=28+28*9/7+28*(9/7)^2=110.2
• 10 years agoHelpfull: Yes(7) No(6)
• As per logarithmic properties:
  Logab =(logb/loga) _________________________(1)
  Loga+logb+logc=log(abc) ___________________(2)
  Logam=n then m=an _____________________________________(3)
  As per progression condition:
  If a, b, and c are in G.P. then b2=ac _____________(4)
  Above following properties have been used to demystify the problem.
  (loga+logb+logc)/log6=6
  Now:
  Log(abc)/log6=6
  abc = 66 (rule 1st and 3rd )
  abc= 46656--------------------------------(5)
  as it is given:
  b2=ac (in the question) a,b,c are in gp (rule4)
  hence putting it in 5th:
  we get b3 = 46656
  hence:
  b=36
  it is also given b-a cube should be perfect square:
  that implies: a=9 as (b-a) =27 that is perfect cube of 3.
  Hence a: 9, b=36
  Then b2=ac from here:
  C= b2 / a;
  C=36x36/9
  C=144;
  So finally we get:
  a = 9
  b = 36
  c = 144
  hence a+b+c = 189
  Answer: 189
• 8 years agoHelpfull: Yes(1) No(0)
• there should be log (base 6) abc=6
  abc=6^6
• 9 years agoHelpfull: Yes(0) No(4)