Problem 149 - College Algebra

Question

Research applications of logarithmic functions as mathematical models and plan a seminar based on your group's research. Each group member should research one of the following areas or any other area of interest: \(\mathrm{pH}\) (acidity of solutions), intensity of sound (decibels), brightness of stars, human memory, progress over time in a sport, profit over time. For the area that you select, explain how logarithmic functions are used and provide examples.

Step-by-Step Solution

Answer

Logarithmic functions are used in the measurement of sound intensity, which is measured in decibels. A tenfold increase in the intensity of a sound corresponds to an increase of 10 on the decibel scale, demonstrating the logarithmic relationship between intensity and decibel level.

Step 1: Understanding the use of logarithms in sound intensity

Sound intensity is measured in decibels, a logarithmic scale. Decibels are used because the human ear's perception of sound is logarithmic, not linear. Basically, a small increase in decibel level equates to a large increase in actual sound intensity. As a general rule, an increase of 10 decibels is perceived as roughly doubling the volume of sound. Mathematically, the formula to compute sound level L in decibels, given an intensity \(I\) and a reference intensity \(I_{0}\), is \(L = 10 \cdot log_{10}(\frac{I}{I_{0}})\).

Step 2: Giving an example of the logarithmic nature of decibels

Let's walk through a practical example. Imagine we have two sounds, one with an intensity of \(1 \times 10^{-12}W/m^2\) (the approximate threshold of human hearing, we will call this Sound A), and another with an intensity of \(1 \times 10^{-11}W/m^2\) (Sound B). Using the formula from Step 1, the decibel level of Sound A would be \(10 \cdot log_{10}(\frac{1 \times 10^{-12}}{1 \times 10^{-12}}) = 0 dB\). Similarly, the decibel level of Sound B would be \(10 \cdot log_{10}(\frac{1 \times 10^{-11}}{1 \times 10^{-12}}) = 10 dB\). This demonstrates that a tenfold increase in the intensity of sound (from \(1 \times 10^{-12}W/m^2\) for Sound A to \(1 \times 10^{-11}W/m^2\) for Sound B) results in an increase of 10 decibels (from 0 dB for Sound A to 10 dB for Sound B), showing the logarithmic nature of the decibel scale.