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- [Voiceover] All right,
I'm going to tell you

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about the Decibel Scale.

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This is the scale that we use

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to figure out the loudness of a sound.

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The equation that goes along
with it looks like this.

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Beta equals 10 log, or
logarithm, base 10 of I

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divided by 10 to the negative
12 watts per square meter.

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This looks intimidating.

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Let's talk about it and break it down.

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Beta is the number of decibels.

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So this side gives you
the number of decibels

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and we abbreviate decibel
with a little d, capital B.

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This is the number of dBs or decibel.

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You've probably seen this
number on your stereo.

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Somewhere reader just in volume

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because we're going to
measure volume and Decibels.

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10, this 10 just denotes
the fact that this is

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the Deci-bel Scale and
not just the bel scale.

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If you didn't multiply by
10, you'd have the bel scale

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but this is multiplied by 10.

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We like the 10.

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We're going to call is Decibel.

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Log, we'll talk about log in a minute.

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Logarithm here.

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I is the intensity of the sound wave.

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So this is the intensity.

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In Physics, intensity is defined

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to be the power divided by the area.

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What this means, think about it this way.

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You got your ear and a sound wave,

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say, is coming toward your ear.

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If you imagine, one.

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Power's in, what?

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Power measure going to watts.

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Area's measure going to square meter.

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So think about intensity this way.

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If you have one square meter.

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Imagine one square meter of area here.

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This doesn't have to be
an actual physical object.

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Just imagine a square meter of area.

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The power that passes through

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that area will be, how many joules?

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If you figure you how many joules

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pass through this one square meter.

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If you asked how many joules per second

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pass through that one square meter.

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How many joulels of sound energy per second

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pass through the one square meter?

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That would be the number of watts

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per meter squared which
would be the intensity.

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So watts is joules per second.

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This gives you an idea of how much energy

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per second pass through
a certain amount of area.

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This part of the equation is my favorite.

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This is my all-time favorite right here.

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This number.

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This 10 to the negative
12 watts per square meter

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represents the threshold of human hearing.

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What that means is this is the softest

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possible sound you can hear.

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Any sound with an
intensity less than that.

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You won't even notice.

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But if it's anything bigger
than that, a human ear

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that's healthy should
be able to detect it.

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Here is why I like this number.

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This is unbelievably small.

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This is one trillion of
a watt per meter square.

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A trillion.

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What this says is that
even if only one trillionth

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of a joule per second passes
through the square meter,

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your ear would still be able
to detect the sound that soft.

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If that doesn't impress you,
let me put it to you this way.

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Imagine we did have one watt.

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Let me put it to you this way.

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If you have one watt, how big of an area?

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A watt isn't really that much.

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A watt is not a lot of power.

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If you have one watt, how
big could you make this area?

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How, spread out.

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How deluded could one watt be spread over?

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How large of an area could
this one watt be spread over

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and still be intense enough
for the human ear to hear it?

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What do you think?

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Football field?

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I don't know, a city?

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No, it turns out.

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If you do the calculation,
I suggest you do it.

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It's interesting.

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You would get, that you can
spread one watt over the entire

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land area of Germany,
about three times over,

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and still it's intense enough
for the human ear to hear.

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That's how unbelievably
sensitive our ears are.

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It's actually...

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I told you it's unbelievable.

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I can hardly believe it myself.

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Let's come back over to here.

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So here's our equation.

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This is the Decibel Scale.

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Why log?

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That's what you're thinking.

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"Why in God's name did the physicist have

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"to put logarithm in here?

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"This scare me."

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This used to scare me too.

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Well, I'll show you why.

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Here's the problem.

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The fact that we can
hear such a soft sound,

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10 to the negative 12
watts per meter squared,

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there's a huge range of human hearing.

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This means we can hear from 10

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to the negative 12 watts per square meter.

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This is your point zero, zero,

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say three, four, five,
six, seven, eight, nine,

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10, 11, with a one watts per square meter

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all the way where there's no upper limit.

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It just blow out your ears.

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But once you get to about one watt

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per square meter, this
one will start hurting.

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This is painful.

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You're not going to be happy over here.

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You're just going to start hurting.

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You'll start getting
hearing losses and not good.

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So it's a huge range.

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12 orders of magnitude.

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This one watt per square meter is

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a trillion times bigger than this side.

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This scale is just way too big.

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This is awkward.

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We want to scale that's small

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or maybe like one to
100 to measure loudness.

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We don't want to measure from one

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to a trillion or a trillion to one.

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That's what log's going to do.

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Logs are great.

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This is a trick for this use.

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This is why I love this trick.

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Logarithms take really big
or really small numbers

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and turn them into nice numbers.

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That's why we're going
to use the logarithm.

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Let me show you what I mean.

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Logarithm, if you don't remember,

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here's what a logarithm does.

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Log base 10 of a number
equals, here's what it does.

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I'm going to stick a number in here.

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Let's stick 100000.

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What log does, log is a curious guy.

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Log is always asking a question.

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Log always wants to know,
okay, if I'm log base 10,

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log wants to know what
number would I raise 10 to

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in order to get this number in here.

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So log looks at this
number in the parentheses.

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This entire number here
and asks what number

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should I raise 10 to
in order to get 100000.

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Well, we know the answer to that.

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We should raise 10 to the fifth.

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If I raise 10 to the
fifth, I'll get 100000.

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So if five is the number
I raised 10 to get 100000,

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then that's the answer to this

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that log base 10 of 100000 is five.

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Look what happened.

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Log took a huge number, 100000,
and turned it into five.

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Well, that's outstanding.

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Log can take huge numbers,
turn them into nice numbers.

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The logarithm base 10 of
one billion would be...

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One billion is a big number.

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That's hard to deal with

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but log takes 10 and asks what number

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can I raise 10 to in
order to get a billion.

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I should raise 10 to the
ninth because I got one, two,

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three, four, five, six,
seven, eight, nine zeros here.

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I raised 10 to the ninth
to get this number.

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So the answer to this question
for the logarithm is nine.

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Oops, that's not nine.

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Nine, and that's why logarithms are good.

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Logarithm took this enormous number

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of billion and turned it into nine.

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So logarithms take enormous
scales turn them in nine scales.

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That's why we like this formula

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which is our Decibel
Scale because it takes

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enormous intensities
and small intensities,

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turns them into nice intensities.

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Let me show you an example with
this equation really quick.

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Let's say you're talking to your friend.

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Maybe you're yelling at your friend.

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You guys are having a heated exchange.

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You're yelling.

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He's next to you.

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These are the sound waves coming at him.

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You're yelling with an intensity of,

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say 10 to the negative fifth.

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That doesn't sound like a lot

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but that's actually,
you're pretty upset here.

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That's pretty loud.

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I want to know how many decibels is this.

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How do we figure out the decibels?

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Well, here's what we do.

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We use our formula for decibels.

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Beta, number of decibels, equals 10 log

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base 10 of the intensity over always 10

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to the negative 12 watts per square meter

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because that's the
softest sound we can hear.

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What do I get?

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10 to the negative fifth is my intensity.

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So I plug this into here.

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I'm going to get beta equals
10 times the log base 10

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of 10 to the negative fifth,
because that's my intensity,

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divided by 10 to the negative 12.

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Now these are both watts per square meter.

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So let's cancel.

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Well, what's 10 to the
negative fifth divided by 10

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to the negative 12 turns out
that's 10 to the seventh.

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I end up with 10 log of 10 to the seventh.

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Now I don't like logs.

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I'll be honest.

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They freak me out but
I can even do this one.

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Log of 10 to the seventh.

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Remember what log does.

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It asks what number do I raise 10 to

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in order to get the
thing in the parentheses.

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Well, the number I raise
10 to to get the thing

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in this parentheses, it's
already 10 to the seventh.

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It's already in this form.

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So I've raised 10 to the seventh
to get 10 to the seventh.

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So the answer to log base 10

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of 10 to the seventh is just seven.

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My final answer beta, the loudness,

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the number of decibels
is going to be 10 times

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log of 10 to the seventh was just seven

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because I have to raise 10 to the seventh

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to get 10 to the seventh.

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10 times seven equals 70.

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I'm yelling at 70 decibels.

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I need to calm down.

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My friend's going to
start getting mad at me.

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That's how you figure out
how loud the sound wave is.