To solve this question, we need to apply the properties of logarithms and simplify the expression step by step.

Given expression: log32.log43.log54...log1615

Step 1: Simplify the expression
We know that log(ab) = log(a) + log(b), so we can rewrite the expression as:
log32 + log43 + log54 + ... + log1615

Step 2: Expand the logarithms
We can further expand each logarithm using the property log(xy) = ylog(x):
(log3 + log2) + (log4 + log3) + (log5 + log4) + ... + (log16 + log15)

Step 3: Simplify the expression
Using the property log(a) + log(b) = log(ab), we can simplify the expression as:
log(3*2) + log(4*3) + log(5*4) + ... + log(16*15)

Step 4: Evaluate the expression
Evaluating the expression, we get:
log(6) + log(12) + log(20) + ... + log(240)

Step 5: Apply the property log(a) + log(b) + log(c) = log(abc)
Using the property mentioned above, we can simplify the expression as:
log(6*12*20*...*240)

Step 6: Find the common difference
The numbers 6, 12, 20, ..., 240 form an arithmetic progression, with a common difference of 6.

Step 7: Find the number of terms
To find the number of terms, we need to find the value of 'n' in the arithmetic progression:
240 = 6 + (n-1)*6
240 = 6 + 6n - 6
240 = 6n
n = 40

Step 8: Find the product
Now, we can find the product of the terms in the arithmetic progression:
6*12*18*...*240 = 6^1 * (6+6)^1 * (6+12)^1 * ... * (6+36)^1

Step 9: Simplify the product
We can simplify the product as:
6^40 * (1+2)^1 * (1+4)^1 * ... * (1+36)^1

Step 10: Apply the property (a+b)^n = a^n * b^n
Using the property mentioned above, we can simplify the product as:
6^40 * 2^40 * 4^40 * ... * 36^40

Step 11: Apply the property a^m * a^n = a^(m+n)
Using the property mentioned above, we can simplify the product as:
(6*2*4*...*36)^40

Step 12: Find the value of the product
Evaluating the product, we get:
(6*2*4*...*36)^40 = (518400)^40

Step 13: Evaluate the exponent
Evaluating the exponent, we get:
(518400)^40 = 518400^40

Step 14: Apply the property log(a^b) = b*log(a)
Using the property mentioned above, we can rewrite the expression as:
log(518400)*40

Step 15: