Logarithms in Acoustics: Understanding Sound and Music, Study notes of Physics

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Chapter 5 Homework Exercises 3, 4, 8, 9 1

By comparison, the human eye can identify more than 7 million colors. This corresponds to wavelengths between 400 to 750 nanometers (Rossing and Chiaverina, 1999), and thus, the ratio of frequencies our eyes respond to is about one octave. Chapter 5 - Hearing 5.1: Range of Hearing As the book says, the range of frequencies our ear will respond to is amazing. The intensity ratio of sounds that bring pain to the ear and sounds that are just barely audible is more than 10^12! The frequency ratio of the highest to lowest frequencies is nearly 1000! This is more than nine octaves… almost ten octaves (2^9 = 512, 2^10 = 1024). 2

5.7: Logarithms in Sound and Music The book lists four applications in logarithms to acoustics:

1. Decibel levels used to express such things as sound level and amplifier gain are based on logarithms.
2. Frequency response of the ear or audio devices are generally expressed on a compressed or logarithmic scale.
3. The keyboard on a piano or other musical instrument is logarithmic.
4. The musical scale is logarithmic (that is, each step is a certain ratio of frequencies). Thus before we can talk about things such as decibels or sound intensity level (mentioned before), it is useful to expand our mathematical repertoire. By definition: The logarithm to the base y of a number N is the power, x , to which y must be raised in order to equal N. Clear as mud? Well, if such clarity exists, let’s start at the beginning. 4

The term logarithm is just another word for “exponent” or “power” (as is in raised to the power of, not the physics definition of power). Let’s look at two mathematical expressions and define some terms. In the above expressions, y is the base , x is the power or exponent and N is the number generated by taking the base to the power. The equation on the left is read as “the logarithm to the base y of N is x ” whereas the equation on the right is read as “ y to the power of x is N ”. These two equations are inverses of each other… they “undo” each other the same way squares and square roots “undo” each other. To attempt to explain logarithms better, I’m going to start with the equation on the right. However, we need to pick a number for this thing called a “base”. 5 ! log y N = x ! yx^ = N

Bases other than 10 Now to test your understanding, or further clarify logarithms (or both) it will be helpful to break away from the common. Let’s examine what happens if the base is 2 instead of 10. We will start by generating a list of numbers by taking our base to different exponents. 2 -2^ = 2 -1^ = 21 = 22 =

Given the above, find the following: Estimate the following: log 2 1 =? (^) Two to the power of what is one? log 2 0.5 =? Two to the power of what is a half? log 2 4 =? Two to the power of what is four? log 2 0.35 =? log 2 6 =? log 2 1.35 =? (^7)

Bases other than 10 8

Further notes:

1. It is common to shorthand the notation a bit. On your calculator you will more than likely see a log x button. The assumption here is that the base is 10, which is nice because base 10 is very common in acoustics. Actually, anytime the base is not written explicitly, it is assumed to be 10.
2. There are three identities the book mentions to help with calculations involving logarithms (here the base is not listed, but can by any number as long as it is the same on both sides of the equation) a. log ( AB ) = log A + log B b. log ( A/B ) = log A - log B c. log A n^ = n (log A) These identities combined with the table on page 93 (shown below) can greatly reduce your dependence on a calculator when the base is 10. x log(x) x log(x) 1 0.000 6 0. 2 0.301 7 0. 3 0.477 8 0. 4 0.602 9 0. 5 0.699 10 1. 10

Examples: Use the identities and the table to find the following logarithms. log 21 log 0. log 49 a. log ( AB ) = log A + log B b. log ( A/B ) = log A - log B c. log A n^ = n (log A) x log(x) x log(x) 1 0.000 6 0. 2 0.301 7 0. 3 0.477 8 0. 4 0.602 9 0. 5 0.699 10 1. 11

Examples: Use the identities and the table to find the following logarithms. log 64 log 3. log 0. a. log ( AB ) = log A + log B b. log ( A/B ) = log A - log B c. log A n^ = n (log A) x log(x) x log(x) 1 0.000 6 0. 2 0.301 7 0. 3 0.477 8 0. 4 0.602 9 0. 5 0.699 10 1. 13

In addition to the uses previously mentioned, logarithms can be thought of as a way to rescale numbers. This is useful if the quantity involved (such as the frequencies our ears are sensitive to) span many decades (or powers of ten). In situations where this is the case, you will see the scale be logarithmic instead of linear. Linear scale : equal distances represent equal increments Logarithmic scale : is one on which equal distances represent the same factor anywhere on the scale. 14

Other examples of logarithmic “graphs” 16