If you increase the acoustic intensity by 1000 times, how many decibels have you gained?

When discussing changes in acoustic intensity measured in decibels (dB), it's important to remember that the decibel scale is logarithmic. Specifically, an increase of 10 decibels represents a tenfold increase in intensity.

To determine how many decibels correspond to an increase in intensity by 1000 times, we can use the formula for calculating decibels based on intensity:

[ \text{dB} = 10 \log_{10}\left(\frac{I_2}{I_1}\right) ]

where ( I_2 ) is the final intensity and ( I_1 ) is the initial intensity.

In this scenario, if you increase the intensity by 1000 times, the ratio ( \frac{I_2}{I_1} ) equals 1000.

Calculating the decibel increase:

[ \text{dB} = 10 \log_{10}(1000) ]

[ \text{dB} = 10 \log_{10}(10^3) ]

[ \text{dB} = 10 \times 3 ]

[ \text{dB} = 30 ]

Thus,