How to compute the logarithm of a computable number

Question

Let's say you have a computable number $x>0$. $\ln(x)$ is computable as well. Given the computability of $x$, what is a computation for $\ln(x)$. I am using the definition where $a$ is computable if there is a computable function $f_a$ such that $$\frac{f_a(n)-1}n<a<\frac{f_a(n)+1}n$$ so my question becomes given $f_x$ how do you write an algorithm for $f_{\ln(x)}$. If you like though, you can use any definition of computable number will do (so long as you use the same one for $x$ and $\ln(x)$.)