# Taylor Series of a logarithmic function

I was reading Intro to Modern Statistical Mechanics by David Chandler, on page 63. He states the following:

we can expand $$\ln\Omega(E-E_v)$$ in the Taylor series $$\ln\Omega(E-E_v) = \ln\Omega(E) - E_v\frac{d\ln\Omega}{dE} + \cdots$$

Here $$E_v$$ denotes the energy for a particular system state $$v$$.

I was trying to compare this to the Taylor series formula: $$f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)(x-a)^2}{2!}+ \cdots$$

But I am not sure I am following what the author said.

• Note that the Taylor serie for logarithm has very slow convergence. With advantage is used e.g the Padé approximation instead. See www.nezumi.demon.co.uk/consult/logx.htm – Poutnik Mar 25 at 0:18

Remember that $$\Omega$$ is a function of $$E$$ so you have to use the chain rule. The equivalent mathematical formulation is - $$\ln(f(a-x))\approx\ln(f(a))-\underbrace{\frac{f'(a)}{f(a)}}_{\frac{\partial\ln(f(x))}{\partial x}\large|_{x=a}}x+\dots$$
and so on... Use the following command in mathematica/WolfraAlpha to get expansion to order $$n$$ (replace $$n$$ with integer)
Series[Log[f[a - x]], {x, 0, n}]