Does any logarithmic function decreases faster than increases?

Question

I am trying to prove the following:

$$\log_2\left(\frac{f(x)}{f(x-a)}\right)>\log_2\left(\frac{f(x+a)}{f(x)}\right).$$

Is the above true for any positive $f(x), f(x+a)$ and $f(x-a)$?

Answer 1

Since the logarithm is strictly increasing, this inequality is equivalent to

$$\frac{f(x)}{f(x-a)}>\frac{f(x+a)}{f(x)} \quad \iff \quad f(x)^2 >f(x+a)f(x-a)$$

and this is of course not true for all positive $f$ (just take $f\equiv 1$ for example).