Problem 14 - Algebra 2

Question

An equation for loudness \(L\) in decibels, is \(L=10 \log _{10} R\) where \(R\) is the relative intensity of the sound. Solve \(75=10 \log _{10} R\) to find the relative intensity of a concert with a loudness of 75 decibels.

Step-by-Step Solution

Answer

The relative intensity is approximately 3,162,277.66.

Step 1: Understand the Equation

The given equation is \(L=10 \log_{10} R\), which relates loudness \(L\) in decibels to the relative intensity \(R\). You need to solve for \(R\) when \(L = 75\).

Step 2: Substitute Values

Substitute 75 for \(L\) in the equation: \(75 = 10 \log_{10} R\).

Step 3: Isolate Logarithmic Term

Divide both sides of the equation by 10 to isolate the logarithmic term: \(\frac{75}{10} = \log_{10} R\), which simplifies to \(7.5 = \log_{10} R\).

Step 4: Exponentiate to Remove the Logarithm

To solve for \(R\), rewrite the logarithmic equation in its exponential form: \(R = 10^{7.5}\).

Step 5: Calculate the Exponential

Calculate \(10^{7.5}\) to find \(R\). This evaluates to \(R \approx 3,162,277.66\).

Key Concepts

Sound Intensity

Sound intensity is a measure of the power per unit area of a sound wave. It describes how much sound energy passes through a certain area in a given time. This is crucial when considering how loud or quiet a sound is perceived.
Understanding sound intensity allows us to quantify the force of vibrations that reach our ear. The stronger the vibrations, the higher the intensity. It involves the actual physical properties of the sound wave such as:

• The amplitude of the wave
• The distance from the sound source

Generally, when discussing sound intensity, we are talking about how these waves spread out as they travel. As you move further away from a sound source, the energy distributed across a larger area makes the perceived intensity decrease. Hence, sound intensity can also relate to how sound dissipates in an environment.

Decibels

Decibels are units of measurement that express the loudness of a sound. They provide a logarithmic way to describe sound levels, which is more in line with how humans perceive changes in loudness. If you are unfamiliar with logarithms, think of them as a way to transform large numbers into more manageable ones.
The formula to calculate decibels in terms of relative sound intensity is:\[L = 10 \log_{10} R\]

• "\(L\)" stands for loudness in decibels.
• "\(R\)" represents the relative intensity in comparison to a reference level.

Doubling the intensity does not double the decibel level but increases it by approximately 3 dB. This non-linear scale helps when dealing with the wide range of sound levels, from the faintest whisper to a roaring jet engine. Thus, decibels are essential in sound measurement because they align with how our ears perceive differences in sound intensity.

Exponential Equations

Exponential equations are equations where the variables appear as exponents. In the context of logarithmic equations, which are the inverse of exponential equations, they help us solve measures of growth or intensity, like sound.In our earlier example, to find relative intensity, we used the exponential form of a logarithmic equation: By setting \(7.5 = \log_{10} R\), we rewrote it as an exponential equation:\[R = 10^{7.5}\]The logarithmic form helps to solve for an unknown exponent (relative intensity \(R\)). Then we compute the actual value using the exponential equation, finding that \(R \approx 3,162,277.66\), showcasing sound's "growing" effect as perceived through its relative intensity.

• Exponential equations like these are powerful in modeling relationships that change rapidly, such as compound interest or population growth.
• They can appear daunting, but breaking them down through related logarithmic forms makes them manageable and easier to understand.