a,b,c are positive numbers such that they are in increasing Geometric progression then how many such numbers are there in (loga+logb+logc)/6 =log6

• given that a,b,c are in gp
  so b/a=c/b =>b^2=ac
  (loga+logb+logc)/6=log6
  =>log(abc)=6*log6
  =>log(ac*b)=log(6^6)
  =>log(b^3)=log(6^6)
  =>b^3=6^6=>b^3=(6^2)^3
  =>b=6^2=36
  it means a=2,c=18 or a=3 c=12 or a=4 c=9 or a=6 c=6.
  hence there are four such numbers
• 8 years agoHelpfull: Yes(8) No(6)
• (loga+logb+logc)/6=log6
  or, log(abc)=6*log6
  or, log(ac*b)=log(6^6)
  or, log(b^3)=log(6^6)
  or, b^3=6^6
  or, b^3=(36)^3
  or, b=6^2
  or, b=36
  now we got b=36 and because a,b,c are in a G.P
  b^2=a*c
  that means, a*c=6^4.
  so, there are 5 possible values for a,b & c which are:-
  (2,36,648) (3,36,432) (6,36,216) (12,36,108) (18,36,72).
  and also a,b,c can not be (36,36,36) because question already told us that GP is incresing.
  
• 8 years agoHelpfull: Yes(4) No(0)
• in question if it is positive number then infinite number of solution possible but if it is positive integer then 8 solutions are possible
  
  a = a
  b = ar
  c = ar^2
  
  from question
  
  log a + log b + log c = 3log(ar)
  solving the equation given in question we get
  
  log(ar) = log36
  hence ar = 36
  
  so the gp is
  
  1, 36, 1296
  2, 36, 648
  3, 36, 432
  4, 36, 324
  6, 36, 216
  8, 36, 162
  9, 36, 144
  12, 36, 108
  18, 36, 72
  
• 8 years agoHelpfull: Yes(3) No(0)