# SCARCS Calendar:

Today is Tuesday, 2019-11-12

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howto:hambasics:mathbasics

# The decibel

A decibel (dB) is a way of saying how many times bigger (or smaller) something is compared to something else. For example, let's use the faintest sound that the human ear can detect as the basis. A normal conversation would be a million times louder, and a jack hammer would be a hundred billion times louder. To represent these vast differences in numbers, we use a logarithmic scale based on powers of 10.

Here's the idea. Start by representing numbers with their exponents instead:

Number Base 10 Exponent
1 100 0
10 101 1
100 102 2
1000 103 3
10 000 104 4
100 000 105 5
1 000 000 106 6

For now, let's call the exponent a “bel” (not decibel yet), which no one does, but if it's not a thing, it should be.

• Instead of saying that a sound is 1000 times louder, we could say that it's 3 bels louder, and
• instead of saying that it's 10,000 times louder, we could say that it's 4 bels louder.

But what if it's 5000 times louder?

• Since 5000 is between 1000 and 10,000, it might be tempting to say that it would 3.5 bels louder (midway between 3 and 4), but let's check to see if it works: 103.5 = 3162, which is between 1000 and 10,000, but not 5000.
• By trial and error, you might get pretty close: 103.7 = 5011… so 5000 times louder is roughly 3.7 bels.
• But a faster way of finding the exponent of a base 10 number is to use the Log button on your calculator: 36)

But in the same way that milli means a thousandth, and centi means a hundredth (think millimetre and centimetre), a deci means a tenth. So 1 bel contains 10 decibels (like 1 metre contains 10 decimetres, 100 centimetres, or 1000 millimetres).

So another way of saying 3.7 bels is to say 37 decibels, or 37 dB. So here's our table again with a few common “in between” numbers:

Number Base 10 Exponent (bels) decibels (dB)
1 100 0 0
2 100.3 0.3 3
4 100.6 0.6 6
10 101 1 10
100 102 2 20
1000 103 3 30
10 000 104 4 40
100 000 105 5 50
1 000 000 106 6 60
36)
Depending on your calculator, you might have to type it in this order 5000Log 