Chapter 3.6, Problem 65ES

(a) Use a CAS to show that if $k$ is a positive constant, then

$\underset{x\to +\infty }{\lim }x\left(k^{1/x}-1\right)=\ln k$

(b) Confirm this result using L’Hopital’s rule.

(c) If $n$ is a positive integer, then it follows from part (a) with $x=n$ that the approximation

$n\left(\sqrt[n]{k}-1\right)\approx \ln k$

should be good when $n$ is large. Use this result and the square root key on a calculator to approximate the values of $\ln 0.3$ and $\ln 2$ with $n=1024$ , then compare the values obtained with values of the logarithms generated directly from the calculator.