Taylor series expansion of log function.

Question

Hi please help me with Taylor series expansion of $\log\{\frac{a*(1-x)}{x}\}$ upto order=3.

Answer 1

Hint: the Taylor expansion formula with Lagrangian remainder is: $$\frac {d^{n+1}f}{(n+1)!df^{n+1}} \vert _{x=\zeta}(x-x_0)^{n+1}+\sum _{i=0} ^n \frac {d^if}{i!df^i} \vert_{x=x_0}(x-x_0)^i$$ where $\zeta$ is a value between $x_0$ and $x$.

Answer 2

This function has no Taylor expansion around $x_0=0$ because $f(0)$ is not defined.