Problem 85 - Precalculus with Limits

Question

In Exercises \(85-88,\) use the following information. The relationship between the number of decibels \(\beta\) and the intensity of a sound I in watts per square meter is given by $$ \boldsymbol{\beta}=10 \log \left(\frac{I}{10^{-12}}\right) $$ Use the properties of logarithms to write the formula in simpler form, and determine the number of decibels of a sound with an intensity of \(10^{-6}\) watt per square meter.

Step-by-Step Solution

Answer

The sound with an intensity of \(10^{-6}\) watt per square meter has a decibel level of 60 decibels.

Step 1: Simplify the Decibel Formula

Start by using the property of the logarithm, where \(\log(a / b) = \log(a) - \log(b)\). Thus, \(\beta = 10 \log(I) - 10 \log(10^{-12})\). Due to another property of logarithms, \(10 \log(a) = \log(a^{10})\), so the formula now simplifies to \(\beta = \log(I^{10}) - \log((10^{-12})^{10})\), further simplifying to \(\beta = \log(I^{10}) - \log(10^{-120})\).

Step 2: Further Simplify the Formula

Use the property of logarithms \(\log_a a = 1\). In this case, the second part of the expression, \(\log(10^{-120})\), reduces to -120 because \(log_{10}(10) = 1\), so we have \(\beta = \log(I^{10}) + 120\).

Step 3: Calculate the Decibel Level

Use the simplified formula to find the decibel level when the sound intensity is \(10^{-6}\) watts per square meter. Substitute \(I = 10^{-6}\) in the simplified formula. We end up with \(\beta = \log((10^{-6})^{10}) + 120 = \log(10^{-60}) + 120 = -60 + 120 = 60\) decibels.

Key Concepts

Logarithmic Properties

Logarithmic properties are essential for simplifying complex expressions involving powers and roots. One fundamental property is that

• \( \log(a / b) = \log(a) - \log(b) \)
• which means that the logarithm of a division can be split into the subtraction of the logarithms.

This can greatly simplify calculations, especially in sound intensity problems.
Moreover, the power property of logarithms states that \( \log(a^n) = n \cdot \log(a) \). Understanding these properties allows one to manipulate and simplify expressions easily.
For instance, in the given decibel problem, using \( \log(10^{-12}) = -12 \cdot \log(10) \) and since \( \log_{10}(10) = 1 \), we reduce the expression effortlessly. The beauty of logarithms is their ability to turn multipliers into additives, streamlining computations significantly.

Sound Intensity

Sound intensity refers to the power carried by sound waves per unit area in a direction perpendicular to that area. This measurement is expressed in watts per square meter (W/m²),

• indicating how much sound energy reaches a given area.
• Sound intensity plays a vital role in quantifying how "loud" a sound is perceived to be.

The decibel scale is used to express sound intensity in a more manageable and understandable format.
The formula \( \boldsymbol{\beta}=10 \log \left(\frac{I}{10^{-12}}\right) \) connects the sound intensity directly to the perceived loudness or decibels.The reference intensity, \(10^{-12}\) W/m², approximates the faintest sound that the average human ear can detect. As such, understanding sound intensity and its logarithmic relationship allows us to evaluate and compare sound levels efficiently.

Watts Per Square Meter

In physics and acoustics, measuring intensity in watts per square meter (W/m²) provides a consistent method to quantify sound power. Intensity in W/m²

• describes the sound's energy as it propagates through an area.
• This measure is crucial as it directly correlates with the energy that our ears perceive as sound.

Decibels express this metric on a logarithmic scale to make the numbers manageable. The relationship described by the formula \( \beta=10 \log \left(\frac{I}{10^{-12}}\right) \) simplifies complexity but maintains the integrity of measurement.
When converting watts per square meter to decibels, putting the intensity into this formula factors in how human hearing responds to different sound levels. By making these calculations, sound intensities become more applicable in practical scenarios.

Simplification Process

The simplification process involves using logarithmic properties to make equations more manageable. Initially,

• \( \beta = 10 \log(I) - 10 \log(10^{-12}) \) simplifies the base formula by breaking it into more accessible parts.
• By recognizing that \( \log(10^{-12}) \) converts into \(-12\) due to logarithmic properties, we unpack the formula further.

We also employ the property \( 10 \log(a) = \log(a^{10}) \), translating the formula to \( \beta = \log(I^{10}) + 120 \).
In our exercise, once substituted into the formula with \( I = 10^{-6} \), we find that it simplifies neatly to \( \beta = -60 + 120 \) leading to a concise solution of 60 decibels. The structure and clarity provided by simplification help transform potentially intimidating equations into straightforward calculations.