Question

If x = a log 0.001 to the base 0.1, y = log 81 to the base 9.  Then, what is the value of √x-2√y

Answer 1

a LOG X = LOG X^a
LOGa Y  = LOGb Y  * LOGa b         
LOG Y to the base a = LOG Y to the base b * LOG b to the base a
Also,  LOG Y to base X = 1/LOG X to base Y
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x = a LOG 0.001 to the base 0.1
   = a LOG 0.001 to the base 10 * LOG 10 to the base 0.1
   = a LOG 10^-3 to the base 10 * 1/LOG 0.1 to base 10
   = a (-3 LOG 10 base 10) * 1/ LOG 10^-1 to base 10
   = - 3 a / (-1)  LOG 10 base 10 
 x  = 3 a
OR,    (0.1)^x = (0.001)^a   => (10)^-1x = 10^-3a    =>  x = 3a

y = log 81 base 9        =>  9^y = 81        =>  y = 2

So √x - 2√y  =  √(3a) - 2√2