Greenwood press

54
LINEAR FUNCTIONS

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LOGARITHMS
Logarithms are exponents, so they are used to reduce very large values into
smaller, more manageable numbers. It is easier to refer to the number 13.4 than
the number 25,118,900,000,000, which is approximately equal to 10
13.4
. A num-
ber x is said to be the base b logarithm of a number y, if y = b
x
. The correspon-
ding logarithmic equation is x = log
b
y. Base-10 logarithms are used to change
numbers to powers of 10. For example, 500 
≈ 10
2.69897
, so 2.69897 is said to be
the base-10 logarithm of 500. This is commonly written as log 500 
≈ 2.69897.
The decimal part “.69897” is called the mantissa, and the integer part “2” is
called the characteristic. Until inexpensive calculators made it easy to do multi-
plication, division, and roots, scientists and engineers used base-10 logarithms to
simplify computations by changing multiplication of numbers into addition of
exponents, and division of numbers into subtraction of exponents. Up until
twenty years ago, the main computational device for high school students in
advanced math and sciences was based on logarithmic scales—the slide rule.
Other common bases for logarithms are the numbers e and 2. The number e 
≈
2.718281828459. It can be developed from the compound-interest formula as the
limit of (1 + 1/n)
n
as n increases without bound. The base e is used in exponential
expressions that evaluate continuously compounded interest. Logarithms to the
base e are typically written with the abbreviation ln, called a natural logarithm.
ln(500)
≈ 6.21461, because 500 ≈ e
6.21461
. Mathematical functions using e and ln
simplify computations with rates and areas that result from situations in physics,
biology, medicine, and finance. Hence e and natural logarithms are often used in
the statement of rules and properties in these fields. Base-2 logarithms emerge
from the study of computer algorithms. Computers are based on on-off switches,
so using base-2 logarithms provides a natural connection with machine operations.
Logarithmic scales are used in newspapers, households, and automobiles as
well as in scientific research. How loud is a rock concert? Noise is measured in
decibels, a logarithmic scale that is easier to use than the sound-energy measure-
ment of watts per square meter. A decibel is one-tenth of a bel, a unit named after
Alexander Graham Bell (1847–1922), inventor of the telephone. A soft whisper
is 30 decibels. Normal conversation is at 60 decibels. If you are close to the stage
at a rock concert, you hear music at 120 decibels. If you are so close that the
music hurts your ears, the amplifiers are at 130 decibels. Because the decibel
scale is logarithmic, changes along the scale are not linear. When the rock music
moves from very loud (120 decibels) to painful (130 decibels), your ears are
receiving 10 times as much sound energy. The difference of 70 decibels between
normal conversation (60 decibels) and pain (130 decibels) represents 10
7
more
watts per square meter of sound energy.
People’s perceptions of changes in sound intensity are more aligned to the
decibel scale rather than the actual changes in energy level. The same goes for
the perception of light. The brightness of stars was first put on a quantitative
scale by the Greek astronomer Hipparchus at around 130 
BC
. He arranged the vis-
ible stars in order of apparent brightness on a scale that ran from 1 to 6 magni-