Welcome to an application of logarithms.
Sound is measured in a logarithmic scale
using a unit called a decibel.
The decibel equation or formula is shown here
where d equals the number of decibels,
P is the power or intensity of the sound,
and P sub zero is the weakest sound
that the human ear can hear.
The equation is d equals 10 times
log of P divided by P sub zero.
Notice how there's no base on the logarithm,
which means this is common log or log base 10.
Let's look at an example.
Dishwasher A has a noise raiting of 52 decibels.
Dishwasher B has a noise rating of 56 decibels.
How many times more intense is the noise from dishwasher B?
We'll be answering this question using the decibel formula
or decibel equation.
But because we have two noises, and in the equation,
P equals the power or the intensity of the sound,
we will let A equal the intensity of the sound
of dishwasher A and we'll let B equal the intensity
of the sound of dishwasher B.
So using the equation, because dishwasher A has
a noise rating of 52 decibels, we substitute 52 for d,
which gives us the equation 52 equals 10 times log
or common log of P divided by P sub zero,
for dishwasher A is going to be A divided by P sub zero.
And now what we're going to do is solve the equation for A,
where again, A equals the intensity of the sound
of dishwasher A.
So for the first step we divide both sides by 10,
simplifying we have 5.2 equals log or common log
of A divided by P sub zero.
The next step is to write the log equation
as an exponential equation.
And because this is common log, we know this is log base 10,
so applying the definition of a logarithm shown here
at the bottom, 5.2 equals log base 10
of A divided P sub zero is equivalent to 10
raised to the power of 5.2 equals A divided by P sub zero.
And now to solve the equation for A,
we multiply both sides by P sub zero, we have A.
The intensity of the sound of dishwasher A
equals 10 raised to the power of 5.2 times P sub zero.
And now we'll do the same for dishwasher B.
Dishwasher B has a noise rating of 56 decibels.
So now we substitute 56 for d which gives us
the equation 56 equals 10 times the common log of,
instead of P divided by P sub zero, we will have
B divided by P sub zero,
again, because we define B as the intensity of the sound
of dishwasher b.
And now we solve the equation for B.
For the first step we divide both sides by 10,
which gives us the equation, 5.6 equals the common log
of B divided by P sub zero.
Now we write the log equation as an exponential equation.
This is log base 10.
The equivalent exponential equation is 10
raised to the power of 5.6
equals B divided by p sub zero.
To solve for B we multiply both sides by P sub zero
giving us B equals 10 raised to the power of 5.6
times P sub zero.
Now though we have an equation for the intensity
of the sound of dishwasher A and dishwasher B,
we can determine how many times more intense
the noise is from dishwasher B.
To make this comparison, we use division.
We will divide the intensity of the sound
of dishwasher B by the intensity of the sound
of dishwasher A.
This will tell us how many more times intense
the noise is from dishwasher B.
So B divided by A is equal to 10 raised to the power of 5.6
times P sub zero divided by 10 raised to the power of 5.2
times P sub zero.
P sub zero divided by P sub zero is equal to one.
B divided by A is equal to 10 raised to the power of 5.6
divided by 10 raised to the power of 5.2.
And now we'll go to the calculator
to get a decimal approximation for this quotient.
We have 10 raised to the power of 5.6 right arrow,
divided by 10 raised to the power of 5.2, enter.
If we round to two decimal places,
this is approximately 2.51.
So to guess this quotient is equal to 2.51,
this tells us the noise from dishwasher B
is 2.51 times as intense as the noise from dishwasher A.
Let's write this as a complete sentence.
I hope you found this helpful.