### Goal

Understand the dB and where it comes from. Also understand logarithmic and linear scales.

## Number Prefixes

Before we do anything lets get number prefixes out of the way. They are a way of expressing different size numbers. Often we need to talk about numbers like

Number prefixes work by multiplying or dividing by powers of 10. For example

Note that negative exponents are the same thing as dividing. So

Below are listed many of the number prefix and some of this you will see a lot, other will
only come up from time to time. You should be able recognize them and use them! Note that
the symbols are **case sensitive**!

## Power of 10 | ## Prefix name | ## Symbol |
---|---|---|

$10^{18}$ | ## exa | $E$ |

$10^{15}$ | ## peta | $P$ |

$10^{12}$ | ## tera | $T$ |

$10^{9}$ | ## giga | $G$ |

$10^{6}$ | ## mega | $M$ |

$10^{3}$ | ## kilo | $k$ |

$10^{2}$ | ## hecto | $h$ |

$10$ | ## deca | $da$ |

$10^{-1}$ | ## deci | $d$ |

$10^{-2}$ | ## centi | $c$ |

$10^{-3}$ | ## milli | $m$ |

$10^{-6}$ | ## micro | $\mu$ |

$10^{-9}$ | ## nano | $n$ |

$10^{-12}$ | ## pico | $p$ |

$10^{-15}$ | ## femto | $f$ |

$10^{-18}$ | ## atto | $a$ |

## The Decibel

The **decibel** is a math tool that allows us to talk quantities that cover enormous ranges
through the use of a comparison to a reference measure and a logarithm.

We can hear a difference of 1 *to a million*. Could you imagine specifying volume
between 1 to a million? Hence the dB.

The dB (decibel) changes the range by being a **ratio**. The ratio allows us to use a
relative size. It then changes the range by using a logarithm, a multiplicative way of
describing a quantity rather than an additive one.

### Ratios

A **ratio** compares two quantities of the same unit. Ratios are good for showing the relative
size of one quantity vs whatever it is being compared to. In reality our sense of size is always
based on a ratio. A 7ft tall person is only tall because we have something to compare it to.
This sense of "tallness" gives us an intuitive way to understand the unit of feet. If your
used to meters then 7 feet tall may be hard to understand likewise a 2.2 meter tall person
would be harder for someone who is used to feet to make sense of! It will be valuable to develope
an intuition with other units so we can tell when quantities are bigger or smaller than what
we normally expect.

### Example

Say we have a reference of 6 meters. Then suppose we measure 12 meters. The ratio is 2:1 since 12 meters/6 meters gives 2/1. Notice it simplifies to 2 and if we multiple our 6 meters by 2 we get 12 meters! The ratio of 2:1 gives us a sense of size and instead of talking about bigger numbers like 6 and 12 we can talk about 2 or 1.

Also note that a ratio produces a number that has no unit! When we divided meters over
meters they canceled out and left us with a pure number *relative to what we where comparing it to*! (The number in the denominator.) This is why we could simply multiply our resulting
ratio by the reference (the 6 meters from before). It was just 6 meters times 2. Not 2
meters, just 2. This shows the units work out as we expect them to.

We will see that the decibel or dB uses ratios and so it is a pure number. We will have to attach the unit used to produce the dB so we know what produced dB in the first place! This is also why you may find so many types of dB. A dB can be generated from any comparison! Given of course that you also use the logarithm we will get to later.

Before we get into the dB, here is a table showing how the dB can take an incredible range and shrink to a much more reasonable range. In this case from 1 to 1 million all the way down to numbers between 1 and 60!

## Sound Power | ## Bel | ## deciBel |
---|---|---|

$10^{0} \ SP$ | $0\ Bel$ | $0 \ dB$ |

$10^{1} \ SP$ | $1 \ Bel$ | $10 \ dB$ |

$10^{2} \ SP$ | $2 \ Bel$ | $20 \ dB$ |

$10^{3} \ SP$ | $3 \ Bel$ | $30 \ dB$ |

$10^{4} \ SP$ | $4 \ Bel$ | $40 \ dB$ |

$10^{5} \ SP$ | $5 \ Bel$ | $50 \ dB$ |

$10^{6} \ SP$ | $6 \ Bel$ | $60 \ dB$ |

### Bells

The “Bel” is named after Alexander Graham Bell who is credited with inventing the telephone. This is why the B is capital.

Bells as a ratio are often to big. As you saw before, we went from 1 to a million in just
1 to 6. Because of this we split the result into 10ths, hence the "deci"-Bel, *literally a tenth of a bell*. This gives us a more reasonable way to use the "deci"-Bel or dB.

### Example of how we obtain a dB

Say we have a yodeler and we want to know how loud they are.

Loudness is something determined by humans, it's not something you can really measure; *however*, we can measure pressure and then *ask* how a human perceives it. So first we need a
unit of pressure.

A **pascal** is a unit of pressure that is defined as one newton of force per square meter.
To give some perspective 1 newton is about the force it takes to lift an apple (as long as
this apple is close to 102g so an average sized apple). Then to get the pascal you would apply
this force over a square meter (a standard parking spot is about 16.7 square meters, so a 16th-ish
of a parking spot). The pascal is a “small” unit of pressure when compared to other units that
measure pressure.

For us to have a useful volume we need to have a useful reference. In this case we will use the softest pressure a human can hear. We obtain this number by doing experiments. It is found that this value is

Since the top number is Pascals and the bottom number is Pascals the units match and we can instead replace it with a more generic:

Note this more generic version could have been done by any reference! It could be a comparison of meters, one of computer memory size like bytes to bytes, etc... In our case it is one of pressure. Whatever number it produces would be how many times louder it is than the softest sound a human can hear! This is why the dB is amazing. It can work with any ratio.

At this point we apply the

BOOM! We have created a Bel. We could have used any unit with a reference to do this. *It allows us to create a number the represents how it compares to a reference and
scales that comparison to a much smaller range of numbers.*

In this case we used pressure to produce it so this is a

### Now it's to small

We have a new problem however, now our useful range only goes from 0 to about 7! (SPL will be talked about later and later we will extended this range to 14 Bels). This range is to small to be practical. To solve this new issue of a small range we could use decimals like 1.1 or 1.2. But saying something is 4.8 Bels loud is not intuitive. Instead of using decimals we decided to instead use decibels.

This means we are talking about 1/10 of a bel. So, 1.1 Bels becomes 11 decibels. 5.7 Bels becomes 57 decibels ect…. This allows us a range of 0 dB to 60 dB to express the enormous range of our hearing in a way that makes more sense and is relative to us. (Again, later we will see this range extended.)

This gives us the generic form:

## Or

## Which I often will write as:

### Where did the 10 come from?

We now see that a 10 appears on one side but not the other. This is because the “deci”
part represents the 10. In other words if we had 1 Bel, then splitting into 10 we would
have 10 decibels, which form our original Bel. Hence we must multiply the Bels by 10 to
get the number of *deci*bels.

## Bels becomes decibels

## Logs

Now consider this sequence

Here we really have

If we just look at the powers, we have

The powers increase linearly! By the use of pluse 1 or minus 1! Logs try to take advantage of this so that we can use multiplication and division to move around a number line instead of addition and subtraction!

This can offer us the ability to get to very large numbers very efficiently. For example,
to get to 256 would take 256 plus 1’s to do it, and *it would only take 8 “times 2’s” to get the same number of 256*.

### Distance on a line

On a linear number line the distance between 2 and 4 would not be the same as the distance
between 4 and 8. The distance to move from 2 to 4 is only two while to go from 4 to 8
requires us to move 4. This makes sense a lot of the time, but a lot of the time *it doesn't*.

Consider how we hear frequency. 20Hz to 40Hz is one octave, but 10,000Hz to 20,000Hz is also an octave! On a linear number line, the distances would be hugely different! Yet we hear this difference as if they are the same! A single octave! For this reason it can make a lot more sense to use logarithms (the "times" kind of movement) over arithmetic lines, what I have been called the "plus one minus one" kind of movement.

### Base

The base is the "times of" kind of number we choose to use. We could use 4's:

Or something crazy like 56:

Picking different bases causes the numbers to "scrunch up" according to the multiplication. When picking a base to use for the log we want something that matches what we are trying to describe. We hear pitch in a "times 2" kind of way so we pick log base 2 which is written as:

When describing loudness we hear in a "time 10" kind of way so we choose base 10:

10 is such a common base that we usually omit it and assume it is 10 unless otherwise stated:

The different bases "scrunch" up the numbers to fit the multiple we pick, the job of the log is to find our location on the line. For example if we have log base 10 and ask

### Logarithmic, Exponential and Linear

When we move around in a strictly “add 1 or minus 1” kind of way we call it linear, and when we move around in a multiplicative way we are said to move in a logarithmic or exponential way. Exponents are the inverse of logs. Meaning that a log undoes an exponent and vice versa. Hence why we used powers to describe how the logarithmic line works. We were not interested in the base though! We were interested in value of the power! So the log takes the exponent off the base and tells us what it is.

### Example

Say I have

What is

$\log( 1000)$?What is

$\log_{2}( 8)$?

These can be figured out by hand easily but also ensure you can get the answer with your calculator! This way when doing ones that would be hard to do by hand you have more confidence!

### Not Nice Logs

We will use calculators for all log calculations as most logs involve non-integer answers. For example

Decent calculators will be able to calculate any base. However many calculators only support base 10 and the natural log base e. This is fine, in this class we will only ever use base 10.

What is

$\log( 200)$?What is

$\log(30,000,000)$?What is

$\log( 3,000,000)$?Why is the difference between the answer for questions 2 and 3 only 1?

Now hopeful you can see how the log operates and this equation should make much more intuitive sense now.

There are many special math operations we can do with logs, but I will just use them. In derivations you will see them but you are not expected to be able to do the derivation yourself since I don't expect you to know these properties. The derivations are there for those who want to see where the equations come from.

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