ABOUT EXPONENTS, LOGARITHMS AND DECIBELS Derived from Introduction to Technical Mathematics by Peter Kuhfittig David Moulton, October 1990

The following material is derived from several chapters of Introduction to Technical Mathematics by Peter Kuhfittig. It is intended to provide a brief review guide and explanation of some basic material about algebra. If further depth, and or sample problems and examples are needed, we refer you to the tutor for the course, and more complete materials from which this material was drawn on reserve in the library.

About Positive Exponents an = a*a*a*a*... a (n times) (please recall that "*" means multiplied by)

This leads to the expressions: a2*a4 = (a*a)(a*a*a*a) = a(2+4) = a6 and: a7/a4 = (a*a*a*a*a*a*a)/(a*a*a*a) = a(7-11) = a3 and: (a2)3 = (a2*a2*a2) = a(2*3) = a6

The laws of exponents are summarized: If m and n are positive integers, am an = am+n m n m-n a /a = a (where m >n, and a ≠ 0) (am)n = am*n (ab)n = anbn n n n (a/b) = a /b where b ≠ 0)

Adding or subtracting the exponents in an expression is equivalent to multiplying or dividing the numbers that they represent.

Multiplying or dividing exponents is equivalent to raising the numbers that they represent to the power which the product or quotient represents.

A potential problem in working with these expressions arises from the failure to distinguish between (-xy)2 and -(xy)2.

(-xy)2 = (-1xy)(-1xy) = -12x2y2 = 1x2y2 = x2y2 while -(xy)2 = -1(xy)(xy) = -1(x2y2) = -x2y2 so that: 2 2 (-xy) ≠ -(xy) . The even exponent of a negative numbers yields a positive value, while the odd exponent of a negative number yields a negative value.

(-2)4 = (-2)(-2)(-2)(-2) = 4*4 = 16 and (-2)3 = (-2)(-2)(-2) = 4*(-2) = -8

Another necessary concept is that of the radical or root. When a number multiplied by itself yields a given product, it is said to be the square root of that product, so that: 3*3 = 9 and ÷9 = 3.

In the expression ÷n=m, it is assumed that m*m = n, and that the expression could more literally be written 2 ÷n = m.

3 By the same token, ÷27, which is read as "cube root of 27," is the number which, when multiplied by itself three times, equals 27. The nth root of a is expressed

n ÷a n is called the index of the radical and a is called the radicand.

Interestingly, the square root of a given number could be either positive or negative. By common practice, whenever ÷ is used, the positive root is expected. This is called the principal square root.

4 The same convention applies to other even roots. Therefore, ÷256 equals 4, not -4. At the same 3 time, non-even roots of negative numbers will yield negative results: ÷-128 equals 4.

The Laws of radicals are: n n n ÷ab = ÷a * ÷b n n n ÷(a/b) = ( ÷a)/( ÷b) n n n n ( ÷x) = ÷x = x

It is sometimes possible to simplify radicals, by finding the factor within the radicand whose root is most simply expressed:

÷48 = ÷(16*3) = ÷16*÷3 = 4÷3 ÷50 = ÷(25*2) = ÷25*÷2 = 5÷2 We can use one of the laws of radicals to perform an operation called rationalizing the denominator: A fraction such as: 1/÷a can be written without a radical in the denominator. To do this, we multiply both numerator and denominator by ÷a, so that:

(1/÷a)*(÷a/÷a) = ÷a/(÷a)2 = ÷a/a

About Zero and Negative Exponents: If the laws of exponents are applied to the expression: x3/x3 then, x3/x3 = x(3-3) = x0 At the same time: x3/x3 = 1 since any number other than 0 divided by itself equals 1.

Therefore, the law for the zero exponent is: 0 n = 1, so long as n ≠ 0

By the same token: (n3/n5) = n(3-5) = a-2 = 1/n2 leading to the general expression for negative exponents:

-n n x = 1/x , if n ≠ 0

About Scientific Notation: A convenient way to express very large or small numbers, or large ranges of numbers is to use scientific notation.

Such notation follows the convention: n*10k (where n is a number equal to or greater than 1 and less than 10, and k is an integer).

When we treat 10 this way, each increasing positive integer in the exponent represents another 0 to the left of the decimal and each increasing negative integer represents another decimal place to the right of the decimal point.

Therefore, 50,000 = 5*10,000 = 5*10 4 and 55,000 = 5.5*104 By the same token, .00053 = 53/100,000 = 53/105 = 5.3/104 = 5.3*10-4 When multiplying or dividing in scientific notation, the following process is employed: (3*103)*(4*10-2) = (3*4)*(103*10-2) = 12*10(3-2) = 12*10 = 1.2*102 and (2.7*107)/(9*104) = (2.7/9)*(10(7-4)) = .3*103 = 3*102

About fractional exponents: To define n(1/2), we reconsider the expression (am)n = am*n and also (n(1/2))2 = n (1/2) 2 (1/2) 2 If x = n , then x = (n ) , so that x = ÷n Therefore: (1/2) n = ÷n

As a general rule, since, (a(1/n))n = a, 1/n n a = ÷a am/n = (a1/n)m = (am)1/n

The laws for fractional exponents are: 1/n n a = ÷a m/n n m n m a = ÷a = ( ÷a)

Since fractional exponents are an equivalent way of expressing radicals, numerical expressions with factional exponents are evaluated by changing the expression to the radical form. It is usually easier to find the root first and then the power.

For example, 2/3 3 2 2 8 = ( ÷8) = 2 = 4

About logarithms Logarithms were introduced by a Scotsman, John Napier (1550-1617), as a method for speeding up lengthy arithmetic calculations. As all arithmetic was done by hand at the time, logarithms proved to have tremendous value for the scientific community. The slide rule was an important offshoot of logarithms. With the modern calculator, some of the practical advantages of logarithms have declined, but their use in audio, as the basis for the decibel, continues as a fundamental method of expressing the range of amplitudes encountered in audio and acoustics.

Logarithms are derived from exponents. Consider the following expression: 24 = 16 2 is called the base and 4 the exponent, and we say, "Two raised to the fourth power equals 16." If we start with the number 16, then we can say, "16 is written as a power of 2" or "the exponent corresponding to 16 is 4, if 2 is the base." A logarithm is merely an exponent. Using this idea, we could say “the logarithm of 16 is 4, if 2 is the base," or more simply, "the logarithm of 16 to the base 2 is 4."

This can be expressed as:

log2 16 = 4 where log stands for logarithm.

So: 4 2 = 16 and log2 16 = 4 mean the same thing.

At the same time, the quantity from which the logarithm is derived for a given base (16 in this example) is called the antilogarithm, or antilog.

So, in the expression above,

log2 16 = 4 4 is the logarithm, 16 is the antilog and 2 is the base.

It is useful to think of the logarithm as a function that transforms numbers from their linear integer format to an exponential equivalent. To restate from our algebra review, the main purpose of the function concept is to describe an operation on an independent variable that yields a unique value of the dependent variable. The convention for notation of functions is:

if y is a function of x, y = f(x) which is read to mean "y equals f of x" Therefore,

y = logbx can be expressed as f(x) = logbx and x = antilog by can be expressed f(y) = antilog by

In general, y x = b means the same as y = logbx if b is greater than 0 and it does not equal 1.

The expression

y = logbx is read, "y is equal to the logarithm of x to the base b," or "y equals x to the base b." and x = antilog by is read, "x is equal to the antilog of base b raised to the yth," or "x equals b to the yth power." Properties of logarithms: If we recall the laws of exponents: aman = am+n am/an = am-n (am)n = am*n we can restate these as laws of logarithms, so that:

logamn = logam = logan logam - logan = loga(m/y) r logax = r logax

Therefore: -the sum of two logarithms equals the logarithm of the product of their respective antilogs; -the difference between two logarithms equals the logarithm of the quotient of their respective antilogs; -the product of two logarithms equals the logarithm of one of the antilogs raised to the power of the other logarithm: 2 3 log103 times log102 equals log106, or 1,000 , or 100 ;

The quotient of two logarithms equals the logarithm of the radicand of one of the antilogs to the root of the other logarithm: 3 6 2 log106 divided by log103 equals log102, or ÷10 = 10 = 100.

About Common Logarithms Although logarithms were defined for an arbitrary base, in practice only two bases are used, and for our purposes (audio) only one base, 10, will be used. Logarithms referring to base 10 are called common logarithms. If you are working with a so-called scientific pocket calculator, common logarithms are found simply by invoking the LOG function: typically, you enter the number (antilog) for which you wish to find the logarithm and then press the LOG key (some calculators may reverse the order of entry). The resulting number will be the logarithm for that number to the base 10. The process may be simply reversed, if the logarithm is known and the antilog is desired, by entering the log and invoking the antilog function (this may be labeled an INVERSE LOG function, an ANTILOG function, or a 10x function).

To summarize (working in base 10):

- the sum of logs equals the product of the antilogs; - the difference between two logs equals the quotient of one antilog divided by the - the product of two logs is equal to one antilog raised to the power of the other log; - the quotient of one log divided by another is equal to the radicand of one antilog reduced to the root of the other log. For example: 525*42 = 22050 = log 2.72 + log 1.62 = log 4.34 (approximately; values rounded) 525/42 = 12.5 = log 2.72 - log 1.62 = log 1.1 (approximately; values rounded) 5252 = 275625 = log 2.72*2 = log 5.44 (approximately; values rounded) ÷525 = 22.91 = log 2.72/2 = log 1.36 (approximately; values rounded)

About Bels and decibels As part of the growth of telephony, the logarithmic expression miles-of-loss was coined to predict the loss in signal power as a function of transmission over telephone wires. In the early 20th century, this expression was renamed the Bel, after the telephone's inventor, Alexander Graham Bell. The Bel is, in its original usage, simply a logarithm to the base 10, applied to power ratios in electrical audio signals and acoustic intensity levels. In contemporary usage, it is also used (through extended reasoning) to express pressure, voltage, and current ratios, as well as specific values related to given references. The reason that the Bel, and its derivative the decibel, proved to be useful is due to the remarkable range of loudness that the human hearing mechanism is capable of perceiving. Expressed in terms of power or intensity, it is approximately equal to 1 trillion to 1 between the loudest sound we can endure (the threshold of feeling) and the softest sound we can detect (the threshold of hearing). One trillion is 1,000,000,000,000 or 1012 or log 12 or 12 Bels. One Bel is equal to an order of magnitude (one decimal place). Further, the perception mechanism itself is exponential in behavior, so that a given multiple (or proportion) of change in amplitude is perceived as a constant unit, regardless of the actual scale involved. This characteristic of hearing makes the use of logarithmic scaling, such as the Bel and decibel, more appropriate than linear scaling would be. As understanding of the hearing mechanism has grown throughout the 20th Century, the expression of its immense range using a scale of 12 numbers (Bels) proved to be too coarse a value system, although the Bel proved to be perceptually useful in that a change in power of one Bel (i.e. 10:1) is approximately equal to a perceived doubling or halving of apparent loudness, across the entire range of hearing (so that 4 Bels sounds twice as loud as 3 Bels, and 11 Bels sounds twice as loud as 10 Bels), more or less independently of the physical magnitudes they represent. To achieve a more precise expression, the decibel was adopted, whose unit is equal to 1/10th of a Bel. This works out well, because it is a change in magnitude roughly equivalent to the smallest change in magnitude in loudness perceivable by humans (i.e. the just-noticeable- difference, or JND of loudness).

So, if a Bel is equal to a logarithm to the base 10, then a decibel is that logarithm multiplied by 10.

a power ratio of 20,000:1 = log 4.3 = 4.3 Bels = 43 decibels. a power ratio of .0005:1 = log -3.3 = -3.3 Bels = -33 decibels.

Note, also, that the unit decibel (dB) is not a discrete value, but a ratio with the number 1.

The literal formula for its use is:

decibel = 10*log (power 1/power 2) so that, -first we find the ratio of two powers (which is the antilog), -then we find the logarithm of that value, -then we multiply that value by 10.

The use of the decibel to express pressure will be dealt with separately.