if y>1 and x>=y then what is the maximum value of log [root x/y]base x + log[y/x]base y?
a)-0.5
b)0
c)1
d)0.5

• Question should be log [root x/y]base x + log[root y/x]base y
  
  log [root x/y]base x + log[root y/x]base y = 1/2(log [x/y]base x + log[y/x]base y)
  = 1/2(log x(base x) - log y(base x) + log y(base y) - log x(base y))
  =1/2(2- log y(base x) - log x(base y))
  let k= log y(base x) => 1/k = log x(base y)
  
  =1/2(1-[k+1/k]
  as x,y>1 log x(base y) > 0, log y(base x) > 0 => k> 0
  so K+1/k has a minimum value of 2.
  Put it there it becomes 0. which is max value of the equation
  
  0
• 10 years agoHelpfull: Yes(16) No(3)
• ans should be 0.5
• 10 years agoHelpfull: Yes(11) No(5)
• Answer will be 1
  because solving the expression
  last equation becomes 1-(1/2)[log y to the base x - log x to the base y]
  put the value of y=2 and x=2
  we get maximum value as 1
• 10 years agoHelpfull: Yes(8) No(9)
• answer is 0.....
  sorry for the wrong answer
• 10 years agoHelpfull: Yes(8) No(1)
• ans is 0,zero
  log[root x/y]base x+ log[x/y]base y
  1/2(logx base x-logy base x)+logx base y-log y base y
  put x=10,y=10
  so 1/2(1-1)+1-1)=0
• 10 years agoHelpfull: Yes(6) No(3)
• ans should be -0.5
  
• 10 years agoHelpfull: Yes(5) No(1)
• ans will be 1
• 10 years agoHelpfull: Yes(1) No(3)
• 1/2[logx-logy/logx]+[logy-logx/logy]
  1/2[1-logy/logx]+[1-logx/logy]
  put x=y=2
  1/2[1-1]+[1-1]=0
  so,answer is 0.
• 10 years agoHelpfull: Yes(1) No(1)
• 1/2+1/2[logy]-3/2[logx]....final expression with changing base also
  put val x=y=10
  answer is 0.5
• 10 years agoHelpfull: Yes(1) No(1)
• solving the equation you will get 1/2[2-logx^y-logy^x]
  to maximize this calculate the minimum value of these two by putting y=x
  1/2[2-1-1]=0
• 7 years agoHelpfull: Yes(1) No(0)
• log(√x) base x - log y basex + log( y ^2) base y - log x base y
  0.5 +2 -[ log y base x + log x base ] = 0.5+2-2 = 0.5
  min value of x+ 1/x =2
• 10 years agoHelpfull: Yes(0) No(5)
• .5 is the answer
  becos if u take value of x>y then right part of expresion become undefind then left part give the value approxmatily equal to 0.5 and if u check x=y then it gives 0 .then max value is 0.5
• 10 years agoHelpfull: Yes(0) No(2)
• 0
  (1/2)[log y to the base x - log x to the base y]
  since x>=y
  if we put x=y
  ans.=0
  
• 10 years agoHelpfull: Yes(0) No(1)
• =1/2logbasex (x/y) + logbasey (y/x)
  =1/2[logbasex x - logbasex y] +logbasey y - logbasey x
  =1/2[1 - logbasex y] + 1 - logbasey x
  =3/2 -1/2 logbasex y - logbasey x
  now let x=10 and y=10
  =(3/2) - (3/2)
  =0
  So answer is 0
  
  
• 10 years agoHelpfull: Yes(0) No(1)