Problem 85 - College Algebra

Question

The loudness level of a sound, \(D,\) in decibels, is given by the formula $$D=10 \log \left(10^{12} I\right)$$ where I is the intensity of the sound, in watts per meter.\(^{2}\) Decibel levels range from \(0,\) a barely audible sound, to \(160, a\) sound resulting in a ruptured eardrum. Use the formula to solve Exercises. The sound of a blue whale can be heard 500 miles away, reaching an intensity of \(6.3 \times 10^{6}\) watts per meter \(^{2}\). Determine the decibel level of this sound. At close range, can the sound of a blue whale rupture the human eardrum?

Step-by-Step Solution

Answer

The decibel level of the blue whale's sound is 180. Since this decibel level is above 160, which is the threshold for causing a ruptured eardrum, it's safe to state that the sound of a blue whale at close range can rupture a human eardrum.

Step 1: Substitute Given Values into the Decibel Formula

Firstly, plug the value of intensity given, \(6.3 \times 10^{6}\) watts per meter \(^{2}\), into the formula for loudness level in decibels: \[ D = 10 \log(10^{12} \cdot 6.3 \times 10^{6}) \]

Step 2: Simplify the Logarithmic Expression

Now, simplify the expression inside the logarithm using the properties of exponents. Also, remember: when multiplying two numbers with the same base (here we have the base of 10), add their exponents together:\[ D = 10 \log(10^{12 + 6}) \] This calculation simplifies to: \[ D = 10 \log(10^{18}) \]

Step 3: Evaluate the Logarithm

Since the base of the logarithm is 10 and the argument is \(10^{18}\), it means by definition of logarithm it simplifies to the exponent, which is 18:\[ D = 10 \cdot 18 = 180 \]

Key Concepts

Logarithmic Function

A logarithmic function is essentially the inverse of an exponential function. It helps us find the power to which a number, called the base, must be raised to produce a certain value. In the context of sound, we use logarithmic functions to compute the loudness level in decibels. To understand this, imagine how numbers grow exponentially. A logarithm simplifies this growth by finding the original power or exponent of the base.For example, in the formula \(D = 10 \log(10^{n})\), we use a base of 10, which is common. What makes it useful is that for every tenfold increase in intensity, the decibel level goes up by 10 units. This use of the logarithmic function allows us to manage vast differences in sound intensity levels without dealing with unwieldy numbers.

Intensity of Sound

The intensity of sound refers to the power carried by sound waves per unit area, typically measured in watts per meter squared \(\text{W/m}^2\). It is a way to quantify the strength or the loudness of a sound. In the context of decibels, intensity provides the basis for calculating the sound level using the logarithmic scale.The formula given is: \(D=10 \log(10^{12} I)\), where \(I\) is the intensity. For example, for the sound of a blue whale, the intensity reaches \(6.3 \times 10^{6} \text{W/m}^2\), which indicates a very strong sound. When calculating decibels, the intensity is compared to a reference level, which in this formula is \(10^{12}\). This comparison is what allows us to translate intensity into a decibel scale with the help of logarithms, ultimately helping to measure how loud a sound is compared to a very soft one.

Properties of Exponents

Understanding the properties of exponents is crucial when dealing with logarithms and sound intensity calculations. Exponents tell us how many times to multiply a base number. Here are some key properties:

• When multiplying numbers with the same base, you add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• A logarithm of a number in power form simplifies to the exponent: \(\log_{b}(b^x) = x\).
• Raising a base to an exponent and taking its logarithm essentially gives back the exponent itself.

In the solution provided, the multiplication inside the logarithm \(10^{12} \cdot 6.3 \times 10^{6}\) simplifies using the property of exponents to \(10^{18}\). The property used here is that when two powers with the same base are multiplied, we add their exponents. This results in a simpler logarithmic expression, making the process of calculating decibels more straightforward.