Find the exact value

log [√3]81

and...

log [√3]27

log (base√3) 81 =
log (base3^1/2) 3^4 =
[4/(1/2)] log (base 3) 3 = 8


log (base√3) 27 =
log (base3^1/2) 3^3 =
[3/(1/2)] log (base3) 3 = 3/(1/2) = 6

We know :
log (base a^b) (c^d) = (d/b) * log (base a) c

3 is (√3)² and 81 is 3^4, so 81 = (√3)^8
and so log[√3] 81 = 8

similarly, log[√3] 27 = 6

81 = [√3]^x
x= 8

[√3]^x=27

x= 6