The decibel (dB) is a logarithmic unit used to express the relative difference between two quantities, such as sound pressure levels, power levels, or voltage levels. The use of logarithms allows for a more convenient representation of a wide range of values.
The formula for calculating the decibel value in the case of power levels is typically expressed as:
dB = 10 * log10(P₁/P₂)
However, in the case of voltage levels or sound pressure levels, the formula is:
dB = 20 * log10(V₁/V₂) or dB = 20 * log10(P₁/P₂)
The reason for using the logarithm in the formula is to compress the range of values into a more manageable scale. Many natural phenomena, including sound, exhibit a wide range of intensities. The logarithmic scale helps to represent this range effectively and allows for easier comparisons between different values.
By using logarithms, the magnitude of the difference between two values is represented by a more manageable number, making it easier to interpret. The use of logarithms also corresponds to the way our perception of sound intensity and other sensory experiences works. Our perception of loudness is roughly logarithmic, meaning that we perceive a doubling of sound intensity as an increase of about 10 dB.
The factor of 20 in the formula is specific to power ratios. Since power is proportional to the square of voltage or sound pressure, a factor of 10 for the logarithm is insufficient. By using 20 instead, it accounts for the squaring effect and provides the appropriate scaling for power-related measurements.
Overall, the use of logarithms and the 20log() formula in decibel calculations allows for a more concise and practical representation of the relative differences between values across a broad range of intensities.