Problem 76 - Precalculus

Question

We introduce three widely used measurement scales which involve common logarithms: the Richter scale, the decibel scale and the pH scale. The computations involved in all three scales are nearly identical so pay attention to the subtle differences. While the decibel scale can be used in many disciplines, \(^{13}\) we shall restrict our attention to its use in acoustics, specifically its use in measuring the intensity level of sound. \(^{14}\) The Sound Intensity Level \(L\) (measured in decibels) of a sound intensity \(I\) (measured in watts per square meter) is given by $$L(I)=10 \log \left(\frac{I}{10^{-12}}\right)$$ Like the Richter scale, this scale compares \(I\) to baseline: \(10^{-12} \frac{W}{m^{2}}\) is the threshold of human hearing. (a) Compute \(L\left(10^{-6}\right)\). (b) Damage to your hearing can start with short term exposure to sound levels around 115 decibels. What intensity \(I\) is needed to produce this level? (c) Compute \(L(1)\). How does this compare with the threshold of pain which is around 140 decibels?

Step-by-Step Solution

Answer

(a) 60 dB (b) \(I \approx 3.16 \times 10^{-2} \frac{W}{m^2}\) (c) 120 dB, below pain threshold.

Step 1: Understanding the Formula for Sound Intensity Level

The formula given for Sound Intensity Level \(L\) is \[L(I) = 10 \log \left(\frac{I}{10^{-12}}\right)\]. This equation relates the sound intensity \(I\), in watts per square meter, to the intensity level \(L\), measured in decibels (dB). The formula uses the common logarithm (base 10) to compare the intensity of the sound relative to the threshold of human hearing, which is \(10^{-12} \frac{W}{m^2}\).

Step 2: Compute L(10^{-6})

Insert \(I = 10^{-6}\) into the sound intensity level formula: \[L(10^{-6}) = 10 \log \left(\frac{10^{-6}}{10^{-12}}\right)\]. Simplifying the fraction, we get \[L(10^{-6}) = 10 \log (10^{6})\]. Since \(\log (10^{6}) = 6\), the formula gives \[L(10^{-6}) = 10 \times 6 = 60\]. Thus, \(L(10^{-6}) = 60\) decibels.

Step 3: Determine Intensity for L = 115 Decibels

To find the intensity \(I\) that produces 115 decibels, solve the formula \(L(I) = 115\). Using \[115 = 10 \log \left(\frac{I}{10^{-12}}\right)\], divide both sides by 10 to get \[11.5 = \log \left(\frac{I}{10^{-12}}\right)\]. Raise 10 to the power of both sides to solve for \(I\): \[10^{11.5} = \frac{I}{10^{-12}}\]. Multiply both sides by \(10^{-12}\): \[I = 10^{-12} \times 10^{11.5} = 10^{-0.5} = \sqrt{10^{-1}}\]. Therefore, the required intensity \(I \approx 3.16 \times 10^{-2} \frac{W}{m^2}\).

Step 4: Compute L(1)

To find \(L\) when the intensity \(I = 1\), plug \(I = 1\) into the formula: \[L(1) = 10 \log \left(\frac{1}{10^{-12}}\right)\]. Simplifying, we get \[L(1) = 10 \log (10^{12}) = 10 \times 12 = 120\]. Thus, \(L(1) = 120\) decibels.

Step 5: Compare L(1) to the Threshold of Pain

The computed \(L(1) = 120\) decibels is lower than the typical threshold of pain, which is around 140 decibels. This means that an intensity level of 120 decibels is quite loud, but not yet at the level where it would generally cause pain.

Key Concepts

Richter Scale

The Richter Scale is a logarithmic scale used to measure the magnitude of earthquakes. It was developed in 1935 by Charles F. Richter. Unlike linear scales, a logarithmic scale exponentially increases with each unit.

• Each whole number increase in this scale represents a tenfold increase in the measured amplitude of seismic waves.
• This means that a magnitude 5 earthquake has a shaking amplitude 10 times greater than a magnitude 4 earthquake.
• It’s important to note, while the Richter scale is still widely referenced, geologists now often use the moment magnitude scale for more precision, especially for larger earthquakes.

The simple concept of comparing magnitudes using a baseline makes understanding the Richter scale approachable when grasping logarithmic scales. This foundational principle can help you understand similar scales used for measuring other phenomena.

Decibel Scale

The Decibel Scale is essential for measuring sound intensity. Decibels (dB) provide a way to express sound levels logarithmically, making the immense range of human hearing manageable.

• On the decibel scale, zero decibels (0 dB) is near total silence and is the threshold of human hearing.
• Every increase of 10 dB represents a tenfold increase in intensity, and thus appears to double the loudness to human ears.
• The scale is used not only for sound but also in electronic signals and communications.

Decibels provide a practical use of logarithms in managing both small and large sounds, offering a better understanding of everyday acoustic experiences.

pH Scale

The pH Scale is a logarithmic scale used to quantify the acidity or alkalinity of a solution. This scale ranges from 0 to 14.

• A pH of 7 is neutral, representing pure water.
• A pH less than 7 indicates an acidic solution, with a lower pH indicating greater acidity.
• A pH greater than 7 indicates a basic (or alkaline) solution, with higher numbers indicating stronger alkalinity.

Since the pH scale is based on the logarithm of the concentration of hydrogen ions, it simplifies complex chemical potent balances into more comprehensible values. This allows scientists and laymen alike to easily determine the strength and character of a solution.

Sound Intensity Level

Sound Intensity Level measures how intense a sound is at a point in space, generally using the decibel scale. It's important for understanding how sound impacts our environment.

• The formula to calculate it is: \[L(I) = 10 \log \left(\frac{I}{I_0}\right)\]where \(I_0\) is the standard reference intensity level, typically \(10^{-12} \frac{W}{m^2}\).
• This reference level corresponds to the threshold of hearing—the quietest sound that a typical human can hear.
• The intensity level can also help us assess sound exposure risks, showing why some sounds can be harmful.

Understanding this concept helps people protect themselves from potentially damaging noise and appreciate the nuances of their acoustic environment.

Common Logarithms

Common Logarithms use base 10 and are a helpful way to simplify large figures, especially in scientific analyses. They find applications in numerous fields including seismology, acoustics, and chemistry.

• Each increase in a logarithmic value indicates a tenfold increase in the numerical value.
• They facilitate understanding by reducing complex multiplicative relationships to simple additive operations.
• Editions such as logarithmic scales make vast ranges of data manageable and interpretable.

Understanding these logarithms makes it easier to grasp the logarithmic nature of scales like the Richter, decibel, and pH scales, providing powerful tools for measurable comparisons.