ans is 4

Put 4-1/(3sqrt2) as P Then the entire part within the squareroot can be written as P^(1/2)*P^(1/4)*P^(1/8)*P^(1/16)... to infinity This will give P^(1/2+1/4+1/8....) By using GP properties we know that this is P^1 Now just put that value multiply it by 1/3root2 and solve it, I don't have pen and paper in hand rn, I believe it should give the answer straightaway without manipulation.

Edit: on futher tries, this method wouldn’t work.

Take the entire expression under log as x. Now notice the pattern (1/3root2*root(4 - 1/3root2...)

1/3root2 - root(4...) is the "common" part which repeats. So we can simplify the expression x to be

x = 1/3root2 - root(4x)

On solving this we get x=4/9 and log_3/2(4/9) = -2

6-2 =4

i tried putting 4-1/3root2 as x then solving..but got stuck