I know the formula that helps find an approximate value. In this instance it would be like $\ln 1.2 = \ln (1 + 0.2) \approx 0 + 1 \cdot 0.2 = 0.2$.
But I need to find the value more precisely. I calculated, that the value is $0.18232155679$ so I need an accuracy that can determine it to be $0.1823$. How to I do that?
The taylor series for $ln(1+x)$ is an alternating series. So the error is less than the last term dropped. So you have to continue your expansion until you reach a term that has a value smaller than $10^{-4}$. So, you are actually solving $\frac{0.2^{n}} {n}$ such that it is smaller than the error you want.