Solution:

Given, log x (base 8) * log 1/6 (base 8) = 6

We can write log 1/6 (base 8) as log (1/6) (base 8)/log (8) (base 8) = log (1/6) (base 10)/log (8) (base 10)

So, the given equation becomes:

log x (base 8) * log (1/6) (base 10)/log (8) (base 10) = 6

Taking log (1/6) (base 10)/log (8) (base 10) as constant k, we get:

log x (base 8) * k = 6

log x (base 8) = 6/k

Now, we know that log a (base b) = log a (base c)/log b (base c)

Using this property, we can write:

log x (base 10)/log 8 (base 10) = (6/k)/log 8 (base 10)

log x (base 10) = (6/k)*log 8 (base 10) = (6/k)*3

log x (base 10) = 18/k

Now, we can write:

x = 8^(log x (base 8))

x = 10^(log x (base 10)/log 8 (base 10))

x = 10^(18/k)/3

Substituting the value of k in the above equation, we get:

x = 10^(18/(log (1/6) (base 10)/log (8) (base 10)))/3

x = 10^(18/(log (1/6) (base 8)))/3

Hence, the value of x is 8^(10^(27/2)/3).