what is the best estimate of the value of ln 16?\na. 1.20\nb. 2.77\nc. 3.23\nd. 4.00

Explanation:

Step1: Recall the property of natural - logarithm

We know that the natural - logarithm function (y = \ln x) is the inverse of the exponential function (y = e^{x}), and (e\approx2.718). We also know some common values of the natural - logarithm: (\ln e = 1), (\ln e^{2}=2), (\ln e^{3}=3), (\ln e^{4}=4). Since (e^{2}\approx(2.718)^{2}=7.39), (e^{3}\approx(2.718)^{3}=20.09).

Step2: Estimate the value of (\ln16)

Since (e^{2}<16 < e^{3}), (\ln16) is between 2 and 3. We can use the fact that the natural - logarithm function is a continuous function. We know that (\ln x) is increasing. We can also use the linear approximation or just note that (16) is closer to (e^{3}) than to (e^{2}). Let's calculate more precisely. We know that if (y = \ln x), and we want to estimate (\ln16). We know that (\ln16=\ln(2^{4}) = 4\ln2). And (\ln2\approx0.693), so (4\ln2\approx4\times0.693 = 2.772).

Answer:

B. 2.77