To solve this problem, we can use the property of logarithms that states log(a*b) = log(a) + log(b).

First, let's simplify log8 9:
log8 9 = log(9) / log(8)

Next, let's simplify log27 1/6:
log27 1/6 = log(1/6) / log(27)

Now, we can multiply the two expressions together:
log8 9 * log27 1/6 = (log(9) / log(8)) * (log(1/6) / log(27))

Using the property of logarithms mentioned earlier, we can rewrite this as:
= log(9) * log(1/6) / (log(8) * log(27))

Now, we can evaluate each logarithm individually:
log(9) = 2, since 9 = 3^2
log(1/6) = log(1) - log(6) = 0 - log(6)
log(8) = 3, since 8 = 2^3
log(27) = 3, since 27 = 3^3

Substituting these values back into the expression, we get:
= 2 * (0 - log(6)) / (3 * 3)
= -2 * log(6) / 9

Therefore, log8 9 * log27 1/6 = -2 * log(6) / 9.