I am trying to show that the following relation holds:
\begin{equation} \log(1+ax) = log(x) + o(log(x)) \end{equation} as $x\rightarrow \infty$, where $a$ a positive number. I tried using Taylor expansion but I could not come to the results. Any hints would be really helpful! Thanks!
$$\log(1 + ax) = \log\left(x\left(\frac 1x + a\right)\right) = \log x + \log\left(\frac 1x + a\right)$$