To find the value of the given expression, let's break it down step by step:

Step 1: Simplify the expression within the parentheses
a ^ logb/a . logc/b . logd/c . logt/d

Step 2: Simplify each logarithmic term
logb/a can be simplified as logb - loga
logc/b can be simplified as logc - logb
logd/c can be simplified as logd - logc
logt/d can be simplified as logt - logd

Step 3: Substitute the simplified logarithmic terms back into the expression
a ^ (logb - loga) . (logc - logb) . (logd - logc) . (logt - logd)

Step 4: Apply the laws of logarithms
Using the property log(a) - log(b) = log(a/b), we can simplify the expression further:
a ^ log(b/a) . log(c/b) . log(d/c) . log(t/d)
= a ^ (log(b) - log(a)) . (log(c) - log(b)) . (log(d) - log(c)) . (log(t) - log(d))
= a ^ (log(b) - log(a)) . (log(c) - log(b)) . (log(d) - log(c)) . (log(t) - log(d))
= a ^ log(b) . log(c) . log(d) . log(t) / (a . b . c . d)

Step 5: Simplify further
Using the property a ^ log(b) = b, we can simplify the expression even more:
= b . c . d . t / (a . b . c . d)
= t / a

Therefore, the value of the given expression is t / a.

Summary:
The value of the expression a ^ logb/a . logc/b . logd/c . logt/d is t / a.