@ Anshul. To get an appropriate answer, give some background information to your problem?

If you refers to the dB Fast and slow there is no advantage. To use the exponential decay is a consequence that before the digital arrives they were  measured with a condenser, and the condenser has an exponential decay of his charge.

I am not sure I understand your question and more context would indeed be appreciated. However, exponentially decaying phenomena are common in acoustics since the damping behavior of structures is often exponential, and so are evanescent radiations. Perhaps you are referring to reverberation time?

Assuming you are using an exponential time window on the response signal when you do structral testing using the impact method, e.g with a hammer. The objective then is to avoid leakage from truncation of the response signal at the end of the time window. This is the benefit. The drawback is that your estimates of the structural losse´s will be too high due the decay rate added. The same principle applies to reverberation time  testing of rooms analysing the acoustic response from a chirp or burst.

I believe you are asking about the excitation signal Anshul?. If so, you can characterise a system with any type of signal (an impulse, white noise, pink noise, LMS, or sine sweeps (linear or exponential)). Lately the exponential sweep as an excitation signal is the most well-spread method as its signal to noise ratio is very good, and allows for deconvolution of non-linear components, something very useful in systems which contain distortion (almost any real system) and which none of the other signals allows you to do.

The exponential functions are useful not only in the study of acoustical signals but in the study of "linear" systems which are generally considered because these function are not modified by a linear system: if the input of a linear system is an exponential function the output is the same exponential function multiplied by a complex number whose modulus give the "gain" of the system and the phases the phase change due to the system...