Can difference of the $log$ function be approximated?

Question

I am currently trying to optimize a problem.

$$\text{ArgMax}_x \log(1+f_1(x))-\log(1+f_2(x))$$

Due to the fact that $\log (x)$ is a monotonic increasing function, this is equivalent as to

$$\text{ArgMax}_x\ (1+f_1(x))/(1+f_2(x))$$

But my question is, is this equivalent as

$$\text{ArgMax}_x\ (1+f_1(x))-(1+f_2(x))\ ?$$

If it is not equivalent, can it be approximate?