(Probably a better title is needed)
The following statements are given as hypothesis:
1) Suppose we know that a function $e^{f(t)}$ can be expanded in the following way: \begin{equation} e^{f\left(t\right)}=e^{tc}a+e^{tc'}a'+... \end{equation} 2) we know that $c-c'=w$ so that \begin{equation} e^{f\left(t\right)}=e^{tc}\left[a+e^{-tw}a'+...\right] \end{equation} 3) and also that \begin{equation} \lim_{t\rightarrow\infty}e^{f\left(t\right)}=\lim_{t\rightarrow\infty}e^{tc}\left[a+e^{-tw}a'+...\right]=\lim_{t\rightarrow\infty}e^{tc}a \end{equation} then $e^{f\left(t\right)}=e^{tc}\left[a+O\left(e^{-tw}\right)\right]$
Is it in general true or under some conditions of $c,a,f(t)$ that \begin{equation} \lim_{t\rightarrow\infty}\frac{1}{t}\ln\left[e^{f\left(t\right)}\right]=\lim_{t\rightarrow\infty}\frac{1}{t}\ln\left[e^{tc}a\right] \end{equation}?
Hint: If $g$ is continuous and the limit $$\lim_{t\to\infty} h(t)$$ exists, then $$\lim_{t\to\infty} g(h(t)) = g(\lim_{t\to\infty} h(t)).$$