Find the value of (log2 3)•(log3 4) • …• (logn n + 1) • (logn+1 2)?

we use log_a(b) * log_b(c) = log_a(c)

so (log2 3)•(log3 4) • …• (logn n + 1) • (logn+1 2)
= (log_2 4) • …• (logn n + 1) • (logn+1 2)
= log_2 2
= 1

Presuming the inner number are bases, so:

log₂(3) * log₃(4) * log₄(5) * ... * logₐ (a + 1) * logₐ₊₁ (2)

(I changed to "a" from "n", but it's the same problem.)

If we change the bases so all logs have the same base, we get:

log(3) / log(2) * log(4) / log(3) * log(5) / log(4) * ... * log (a + 1) / log(a) * log (2) / log(a + 1)

Now your values can start cancelling out.. log(3) in the first numerator and the second numerator, then log(4), etc. until you end up with:

1 / log(2) * log (2)

Now the log(2)'s cancel out from the first denominator and last numerator, which leaves:

1

The last term doesn't fit the pattern. Is that correct?