How to change volume of a PCM 16 bit signed audio?
Changing the volume of an audio signal must be done by applying a gain (multiplication) - and optionally clipping if your system has a limited dynamic range. This is as simple as that. Applying a non-linear function to an audio signal will cause distortion and add harmonics, and you don't want this to happen - you want to modify the loudness of the signal, not its timbre. [To be fair, there are non-linear processings designed to change the perceived loudness of the signal without affecting the timbre, within a given dynamic range constraint (eg. multiband compression), but it doesn't look like this is what you need.]

Where non-linearity and fancy response curves come to play is when designing a user interface - when deciding on the relationship between the position of the control (knob or slider, whether on a GUI or as physical hardware) and the gain applied to the signal. This is where perception matters, because the users will expect a mapping between the position of the slider and their perception of loudness. Please note that even if the relationship between the position of the volume control and the gain applied to the signal is non-linear, the process of applying the gain to the signal is linear, and non-linearity would be unwanted there!

When it comes to physical volume controls, eg in hifi systems or personal audio players, the relationship between the knob position and the attenuation is closer to an exponential curve, though its shape has been tweaked and is constrained by the manufacturing process - sometimes it's just two or three linear segments. You can find those curves in the datasheets from manufacturers ("A" taper). Mixing console faders usually have their response compressed so that the upper half of their travel covers the useful range of -20 dB..+6dB.

In the software world - at least for music production - it is most common to have volume/gain knobs calibrated in dB. For example, if you have a 100 pixels long volume slider graduated from -48dB to +6dB, the gain applied to the signal would be $10^{\frac{-48 + 54 \frac{x}{100}}{20}}$.