The definition of logarithms states that if log_a(b) = c, then a raised to the power of c equals b, hence a^c = b.

The logarithm of a product is indeed the sum of the logarithms, as per the property log(a*b) = log(a) + log(b).

Since 81 is 3^4, the base of the logarithm is 3.

For any base b > 0, log_b(1) = 0, since b^0 = 1.

log_2(8) = 3 and log_2(4) = 2, thus log_2(8) + log_2(4) = 3 + 2 = 5.

Since 1000 is 10^3, log_10(1000) = 3.

The last statement is incorrect as it does not follow the properties of logarithms.

All the listed properties are correct and fundamental to logarithmic functions.

Using the property of logarithms, log_2(8) + log_2(4) = log_2(8*4) = log_2(32) = log_2(16).

log_5(25) = 2 and log_5(5) = 1, thus log_5(25) - log_5(5) = 2 - 1 = 1.

Since 0.01 is 10^-2, log_10(0.01) = -2.

Logarithm of zero is undefined, hence log_5(0) is the correct answer.