Logarithms are used to measure the “loudness” of sound. Decibels (dB) are logarithmic units used to describe a ratio of two levels of intensity or pressure. The difference between two levels of sound pressure ($P_{1}$ and $P_{2}$) is defined as $10\log(\frac{P_1}{P_2})$ dB. Usually, when decibels are used to describe just one sound, it is assumed that that sound is being compared to a reference level of $20$ micropascals (a measure of pressure).

1. How many decibels correspond to doubling the pressure of a sound?

   Answer (a):

   $10 \text{log}\left(2\right) ≈ 3.0$

2. What is the sound pressure of a noise described as $60$ dB?

   Hint (b):

   $10\cdot \text{log}\left(\frac{P}{20}\right)=60$

   $\text{log}\left(\frac{P}{20}\right)=6$

   $10^6=\frac{P}{20}$

   Answer (b):

   $P = 20 · 10^{6} = 2 · 10^{7}$

3. What does $0$ decibels mean?

   Answer (c):

   The sound ratio is $1$.

4. How many times more pressure is in a sound of $40$ dB than of $20$ dB?

   Answer (d):

   $100$ times more pressure.