Final answer:
1 decibel (dB) equals 1/10th of a bel, which results in the formula 1 dB = 10 × log10(P1/P2) to maintain consistency when converting from Bels to decibels. The prefix 'deci-' means 'one-tenth,' which explains why you multiply the logarithm of the power ratio by 10 in the definition of a decibel.
Step-by-step explanation:
The confusion in understanding the unit of decibel (dB) vs. bel originates from the prefix 'deci-' which means 'one-tenth.' Hence, 1 decibel is 1/10th of a bel. The bel is defined as the logarithm to base 10 of the ratio of two powers, 1 Bel = log10(P1/P2). When we talk about decibels, we are using a smaller unit, and the conversion is such that 1 dB = 1/10 Bel, which then requires multiplication by 10 for the logarithmic ratio to maintain consistency. So, 1 dB = 10 × log10(P1/P2).
Example 17.3 demonstrates this concept by showing that a sound intensity that is twice as intense as another is about 3 dB higher. This uses the properties of logarithms and can be shown by calculating 10 × log10(2/1), which is approximately 3 dB. This illustrates how the sound intensity level is typically measured in decibels rather than just Bels for practical reasons.