Final answer:
To evaluate the expression log₃(16) without a calculator, we can rewrite it as 4log₃(2) and estimate the value of log₃(2) using a calculator. The result is approximately 1.204.
Step-by-step explanation:
To evaluate the expression log₃(16) without a calculator, we need to determine what power we need to raise the base (10 in this case) to get 16. In other words, we need to find x in the equation 10^x = 16. Using the fact that 16 is equal to 2^4, we can rewrite the equation as 10^x = 2^4. Taking the logarithm of both sides, we have x = log₃(2^4). Since the exponent 4 can be brought down as a coefficient according to the logarithm property, we have x = 4log₃(2). Now, we can estimate the value of log₃(2) by rewriting it as log(2)/log(10), where log(2) is the logarithm of 2 to the base 10. Using a calculator, we find that log(2) is approximately 0.301. Thus, x ∼ 4 * 0.301 ≈ 1.204. Therefore, log₃(16) is approximately 1.204.