Problem 37 - Contemporary Precalculus

Question

The perceived loudness \(L\) of a sound of intensity \(I\) is given by \(L=k \cdot \ln I,\) where \(k\) is a certain constant. By how much must the intensity be increased to double the loudness? (That is, what must be done to \(I\) to produce \(2 L ?\) )

Step-by-Step Solution

Answer

Answer: The intensity must be increased by \(\sqrt{I} - I\) to double the perceived loudness.

Step 1: Write down the given equation

We are given the equation for perceived loudness: \(L = k \cdot \ln I\).

Step 2: Write the equation for doubled loudness

We are asked to find the new intensity \(I'\) when the loudness is doubled: \(2L = k \cdot \ln I'\).

Step 3: Divide the second equation by the first equation

Let's divide the second equation by the first equation: \(\frac{2L}{L} = \frac{k \cdot \ln I'}{k \cdot \ln I}\). This simplifies to: \(2 = \frac{\ln I'}{\ln I}\).

Step 4: Apply logarithm properties to solve for \(I'\)

Using the properties of logarithms, we can rewrite the equation as: \(\ln I'^2 = \ln I\). Now we can apply the inverse of the natural logarithm (exponential) to both sides of the equation: \(I'^2 = I\).

Step 5: Solve for \(I'\)

To find the new intensity \(I'\), take the square root of both sides: \(I' = \sqrt{I}\).

Step 6: Calculate the increase in intensity

In order to find the increase in intensity from \(I\) to \(I'\), subtract the original intensity from the new intensity: Increase in intensity = \(I' - I = \sqrt{I} - I\). Therefore, the intensity must be increased by \(\sqrt{I}-I\) to double the perceived loudness.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a mathematical function that helps model exponential growth and decay processes. It's based on the constant \( e \), approximately equal to 2.718. This logarithm is commonly used in calculus and analysis due to its unique properties. It translates multiplicative relationships into additive ones, making calculations simpler in many scientific contexts.

In the context of sound intensity and perceived loudness, the natural logarithm helps us express the relationship between the physical intensity of sound and how loud it actually seems to us. This is given by the formula: \( L = k \cdot \ln I \), where \( L \) is the perceived loudness, \( k \) is a constant, and \( I \) is the intensity of the sound. Using the natural logarithm helps capture the logarithmic nature of human perception.

Properties of Logarithms

Logarithms have a set of properties that make them powerful tools in mathematics, especially helpful for solving equations involving exponentiation. Some key properties include:

• \( \ln(xy) = \ln x + \ln y \)
• \( \ln \left( \frac{x}{y} \right) = \ln x - \ln y \)
• \( \ln(x^a) = a \cdot \ln x \)

In our exercise, these properties are crucial in solving for the new intensity \( I' \) when the loudness is doubled. By using the property \( \ln(x^a) = a \cdot \ln x \), we rewrote the equation \( 2 = \frac{\ln I'}{\ln I} \) as \( \ln(I'^2) = \ln I \). This transformation allowed us to further solve for \( I' \) by taking the inverse, resulting in \( I'^2 = I \).
Understanding these properties dissolves the complexity of working with logarithms, providing a straightforward path to find solutions.

Sound Intensity

Sound intensity is a measure of the power carried by sound waves per unit area in a direction perpendicular to that area. Typically measured in watts per square meter (W/m²), it quantifies how much sound energy is present.

The formula \( L = k \cdot \ln I \) presents the relationship between intensity \( I \) and perceived loudness \( L \).
As intensity increases, the logarithmic nature of the equation shows that perceived loudness doesn't increase as quickly, reflecting how our ears perceive sound. This non-linear relationship makes sense because human hearing is more sensitive to changes in quieter sounds than louder ones.
The main takeaway is that sound intensity is not perceived directly but through a logarithmic scale, clarifying why the equation uses the natural logarithm to model perceived loudness.

Perceived Loudness

Perceived loudness is the sensation of how loud or soft a sound appears to the human ear. Unlike the physical measure of intensity, perceived loudness is subjective and depends on several factors, including the frequency of the sound and the listener's environment.

The exercise illustrates how doubling the perceived loudness \( L \) requires solving for a new intensity \( I' \) in the equation \( 2L = k \cdot \ln I' \). We discovered that the relationship \( I'^2 = I \) results in \( I' = \sqrt{I} \).
Therefore, the intensity must be increased by \( \sqrt{I} - I \) to achieve a doubled loudness. This demonstrates how perceived loudness requires more than simply doubling the physical intensity. It requires an understanding of the logarithmic relationship captured by the natural logarithm, reflecting the complexities of human auditory perception.