Quadrupling the power output from a speaker emitting a single frequency will result in what increase in loudness (in units of dB)

6.02 dB increase  

Explanation:

Let us take the initial power from the speaker P' = P Watt

then, the final power P = 4P Watt

for a given unit area, initial intensity (power per unit area) will be

I' = P Watt/m^2

and the final quadrupled sound will produce a sound intensity of

I = 4P Watt/m^2

Increase in loudness is gotten from the relation

ΔL =  

where

I = final sound intensity

I' = initial sound intensity

imputing values of the intensity into the equation, we have

==>   =   = 6.02 dB increase