Problem 2.12 (Schroeder’s Intro to Thermal Physics)

Problem 2.12

The natural logarithm function, ln, is defined so that $e^{\ln{(x)}}=x$ for any positive number x.
(a) Sketch a graph of the natural logarithm function.
(b) Prove the identities
$\ln{ab} = \ln{a} + \ln{b}$ and $\ln{a^b}=b \ln{a}$ .
(c) Prove that $\dfrac{d}{dx} \ln{x} = \dfrac{1}{x}$
(d) Derive the useful approximation $\ln{(1+x)} \approx x$ ,
which is valid when $|x|<<1$ . Use a calculator to check the accuracy of this approximation for x = 0.1 and x = 0.01.