Problem 13 - 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 \(130=10 \log _{10} R\) to find the relative intensity of a fireworks display with a loudness of 130 decibels.

Step-by-Step Solution

Answer

The relative intensity of the fireworks display is 10,000,000,000,000.

Step 1: Understand the Equation

We are given the equation for loudness: \( L = 10 \log_{10} R \), where \( L \) is the loudness in decibels and \( R \) is the relative intensity of the sound. We need to find \( R \) when \( L = 130 \).

Step 2: Substitute the Given Loudness

Substitute 130 for \( L \) in the equation: \( 130 = 10 \log_{10} R \). Our goal is to solve for \( R \).

Step 3: Isolate the Logarithm

Divide both sides of the equation by 10 to isolate the logarithm: \( 13 = \log_{10} R \).

Step 4: Solve for R Using Exponential Form

Convert the log equation into its exponential form: \( R = 10^{13} \). This conversion uses the property \( \log_{10} a = b \) implies \( a = 10^b \).

Step 5: Calculate the Relative Intensity

Raise 10 to the power of 13 to find \( R \). This results in \( R = 10^{13} \) which equals 10,000,000,000,000.

Key Concepts

Loudness in Decibels

Decibels serve as a unit measuring the loudness of sound. Understanding how loudness is quantified can be quite fascinating. The equation for loudness, given by \( L = 10 \log_{10} R \), helps us understand how different sounds can be compared.

• In this equation, \( L \) represents the loudness in decibels.
• \( R \) is the relative intensity of the sound.

Decibels are logarithmic, which means each step in decibels represents an exponential change in sound intensity. When you encounter a sudden change in decibels, it might seem small, but it's significant in terms of energy.Comparatively, every 10-decibel increase implies the intensity has risen tenfold. Therefore, a sound of 130 decibels is tremendously loud, reflecting a million times the intensity of a 70-decibel sound. Think of this as a whisper versus a roaring plane engine.

Relative Intensity

Relative intensity \( R \) refers to the power of sound compared to a standard reference point. This concept is crucial in understanding how loud a sound can be.Factors that affect relative intensity include:

• Distance from the sound source: The closer you are, the greater the intensity.
• The power of the sound source: A powerful explosion or fireworks will have a higher intensity.

In our exercise, when finding \( R \), we interprete what a loudness of 130 decibels signifies in power terms. Although an abstract concept, it becomes concrete when converted into an actual number using logarithmic calculations. The relative intensity is critical for situations where managing sound levels is necessary, such as regulating noise pollution or ensuring safety at loud events.

Exponential Form

The exponential form is a way of expressing equations using exponents. It's particularly useful when dealing with logarithmic equations, like those involving sound intensity level. After isolating the logarithm in our equation \(13 = \log_{10} R\), we converted it to exponential form: \(R = 10^{13}\). Here's a quick guide on how this works:

• The base of the log (\(10\) in this case) becomes the base of the exponential form.
• The result of the logarithm is the exponent (\(13\) here).
• "\( R \) equals the base raised to the power of the exponent" defines your exponential expression.

Converting logs to exponents might initially seem tricky, but it’s very logical and offers a powerful means to switch between logarithmic and linear growth, making it essential in fields ranging from sound waves to financial growth modeling.

Solving Equations

Solving equations, especially involving logs, involves a systematic approach.To solve \(130=10\log_{10}R\), we undertake a series of logical steps:

• Start by substituting known values (here, replacing \(L\) with \(130\)).
• Then, carefully isolate the variable or expression with logs, often involving simple algebra like division.
• Once the logarithm is alone, change it to an exponential form to solve for \(R\).
• Finally, calculate to find the numerical solution, knowing each step narrows to your answer.

This process not only solves our given problem, but also acts as a blueprint for tackling complex equations in future. Mastering this strategy of isolating log expressions and converting to exponential form empowers you to solve a range of real-world problems.