Final answer:
The expression log base3 81 equals 4 because 81 is 3 raised to the 4th power, and thus the logarithm is asking for the exponent that 3 is raised to, to get 81.
Step-by-step explanation:
To evaluate the expression log base3 81 without using a calculator, you need to recognize that 81 is a perfect power of 3. In particular, 81 is 3 raised to the 4th power (34 = 81). Because logarithms ask the question "To what exponent must we raise the base to get the number?", the logarithm of 81 with base 3 is simply the exponent we raised 3 to reach 81.
So, here we have log381 = log3(34). According to the properties of logarithms, the log base3 of 34 is 4, because the base of the logarithm and the base of the exponentiation are the same.
Thus, log381 equals 4, which makes option C) 4 the correct answer.