how can I use taylor series to approximate these two functions?

Solution 1:

Assuming that $x$ tends to zero,

1. You can write $\log(x^5) = 5 \log(x)$. This cannot be approximated by polynomials or rational functions of $x$, but such approximation is not needed. One uses results such as $x^\alpha |\log(x)|^\beta \to 0$, for $\alpha > 0$.

2. One can write $\log(2 + e^x) = \log(3) + \log(1 + \frac{e^x-1}{3})$, which has the form $\log(1 + u)$ with $u\to 0$.

In general, if you have $\log(\varphi(x))$ and $\varphi(x)\to \ell > 0$, you can write \begin{equation} \log(\varphi(x)) = \log(\ell) + \log\left(1 + \frac{\varphi(x)-\ell}{\ell}\right) \end{equation} which has the form $\log(1 + u)$ with $u\to 0$.