Question

if x= log 0.001 base 0.1 and y = log 81 base 9 then (x-2(y)^1/2)^1/2 =

a) 3 - √2
b) √3 - 2
c) √2 - 1
d) √2 - 2

Answer 1

First of all I list the basic logarithmic properties to be used in solving the given question:-
1. $log_{x}x = 1$
2. $log_{x} y^{z} = z log_{x} y$ 

x = $log_{0.1}0.001 = log_{0.1} 0.1^{3} = 3$
y = $log_{9}81 = log_{9} 9^{2} = 2$
then, $\sqrt{x-2 \sqrt{y} }$ = $\sqrt{3-2 \sqrt{2} }$
we can compare $\sqrt{3-2 \sqrt{2} }$ as $\sqrt{1+ \sqrt{2} + 2 \sqrt{2} } = \sqrt{(1+ \sqrt{2} )^{2} } = 1+ \sqrt{2}$

Hence, the value of $\sqrt{x-2 \sqrt{y} }$ is $1+\sqrt{2}$

Please tell me if this is the right answer, or in case of doubt, please comment.
:-)