THe loudness L of a sound is measured by L=10 log10^R where R is the sound's relative intensity. One person talks with a relative intensity of 10^6 or 60 decibels. How much louder would 100 people be ,talking with the same intensity.
So would I just use L= 10log10^1/10^6? and then multiply by 100 when done. I am supposed to use the Quotient property of logarithms.
To solve this problem using the Quotient property of logarithms, we need to calculate the relative intensity for 100 people and then use it to find the loudness.
Given that one person has a relative intensity of 10^6, we can calculate the relative intensity for 100 people by multiplying it by 100:
Relative intensity for 100 people = 10^6 * 100 = 10^8
Now, we can use the formula for loudness:
L = 10 * log10(R)
Substituting the calculated relative intensity for 100 people:
L = 10 * log10(10^8)
Applying the Quotient property of logarithms, which states that log(a/b) = log(a) - log(b):
L = 10 * (log10(10^8))
Using the fact that log(b^c) = c * log(b):
L = 10 * (8 * log10(10))
Since log10(10) = 1, the equation simplifies to:
L = 10 * 8
L = 80 decibels
Therefore, 100 people talking with the same intensity would be 80 decibels louder than one person.
To determine how much louder 100 people would be talking with the same intensity, we need to use the given formula L = 10 log10(R), where L is the loudness and R is the sound's relative intensity.
First, let's find the loudness of one person talking with a relative intensity of 10^6 or 60 decibels.
L1 = 10 log10(10^6)
To simplify this expression using the Quotient Property of logarithms, we can rewrite it as:
L1 = 10 [log10(10^6) - log10(1)]
Since log10(10^6) = 6 (logarithm property loga(ax) = x), we have:
L1 = 10 [6 - log10(1)]
The logarithm of 1 to any base is always 0, so log10(1) = 0.
L1 = 10 [6 - 0]
L1 = 10 × 6
L1 = 60 decibels
Therefore, one person talking with a relative intensity of 10^6 or 60 decibels is already producing a sound of 60 decibels.
Now, let's find the loudness when 100 people talk with the same intensity. We can use the property that the total sound intensity is proportional to the number of people.
Since the intensity is the same for each person, the total relative intensity when 100 people talk would be:
R_total = 100 × R (where R is the sound's relative intensity for one person)
Thus, the loudness when 100 people talk with the same intensity would be:
L2 = 10 log10(R_total)
= 10 log10(100 × R)
= 10(log10(100) + log10(R))
= 10(2 + log10(R))
Using the given relative intensity R = 10^6, we have:
L2 = 10(2 + log10(10^6))
= 10(2 + 6)
= 10 × 8
= 80 decibels
Therefore, 100 people talking with the same intensity as one person would be 80 decibels louder.
I1 = 10^6.
I2 = 10^2*10^6 = 10^8.
db(increase) = 10*Log(I2/I1).
db(increase) = 10*Log(10^8/10^6) = 20.