Problem 103 - College Algebra Essentials

Question

The loudness level of a sound can be expressed by comparing the sound's intensity to the intensity of a sound barely audible to the human ear. The formula $$D=10\left(\log I-\log I_{0}\right)$$ describes the loudness level of a sound, \(D\), in decibels, where \(I\) is the intensity of the sound, in watts per meter \(^{2},\) and \(I_{0}\) is the intensity of a sound barely audible to the human ear. a. Express the formula so that the expression in parentheses is written as a single logarithm. b. Use the form of the formula from part (a) to answer this question: If a sound has an intensity 100 times the intensity of a softer sound, how much larger on the decibel scale is the loudness level of the more intense sound?

Step-by-Step Solution

Answer

a. The expression inside the parenthesis written as a single logarithm is \(D = 10*\log(\frac{I}{I_0})\). b. The more intense sound is 20 decibels louder.

Step 1: Simplify the expression inside the parenthesis

By applying the rule \(\log(a/b) = \log a - \log b\), the expression inside the parenthesis can be rewritten as a single logarithm: \(D = 10*\log(\frac{I}{I_0})\).

Step 2: Apply the property of logarithms to find the difference in decibels

To find the difference in decibels of the sound intensity, we need to use the property of logarithms \(\log a^n=n\log a\) for a sound that is 100 times more intense than another. So the difference in decibels can be written as \(D = 10*\log(100) = 10*2 =20\) decibels. The more intense sound is 20 decibels louder.

Key Concepts

Decibel scale

The decibel scale is a logarithmic unit used to measure the intensity of sounds. Unlike linear scales, the decibel scale relates sound intensity to a reference level, allowing for a more practical comparison of very intense and very quiet sounds. The formula used is:

\[ D = 10 \times \log \left(\frac{I}{I_0}\right) \]where:

• \( D \) is the loudness level in decibels.
• \( I \) is the sound intensity in watts per meter squared.
• \( I_0 \) is the reference intensity, typically the intensity of the quietest sound a human ear can detect (usually \( 10^{-12} \) watts per meter squared).

This scale is logarithmic, meaning each increase of 10 decibels represents a tenfold increase in intensity. This approach allows us to handle the vast range of sound intensities we experience daily, from rustling leaves to jet engines. Understanding the decibel scale is essential for fields like acoustics, audio engineering, and hearing science.

Sound intensity

Sound intensity refers to the amount of sound power per unit area. It's measured in watts per meter squared \((W/m^2)\). This concept is crucial in determining how loud a sound is perceived.

Unlike loudness, which is subjective, intensity is an objective measure. When sound waves travel through the air, their intensity decreases with distance due to spreading and absorption. Whether you're evaluating a gentle whisper or a loud concert, the intensity gives a clear picture of sound energy in a given area.
When comparing sounds, the ratio of their intensities is often used. If one sound is 100 times more intense than another, it illustrates a significant difference, reflected clearly on the decibel scale. This comparison is fundamental in applications like noise pollution monitoring and audio equipment testing.

Logarithmic properties

Logarithms play a crucial role in simplifying and solving problems involving exponential growth or decay, like sound intensity. One key property used in manipulating the decibel formula is:

\[ \log \left(\frac{a}{b}\right) = \log a - \log b \]This property allows us to express the ratio of two values as a single logarithm, making calculations more straightforward. For example, the comparison of sound intensities becomes manageable with this transformation, enabling us to express sound levels in decibels.

Logarithms also help us understand multiplicative changes. The property:
\[ \log a^n = n \log a \]means that if a sound's intensity is multiplied by a factor of 100, its decibel level increases by 20, as demonstrated in the original problem. These properties are instrumental in not just acoustics but in all fields dealing with exponential or multiplicative relationships.