Greenwood press

56
LOGARITHMS
sound intensity 
relative intensity
(watts per square meter)
(decibels)
10
3
Military rifle
150
10
2
Jet plane (30 meters away)
140
10
1
Pain level
130
10
0
Amplified rock music
120
10
-1
Power tools
110
10
-2
Noisy kitchen
100
10
-3
Heavy traffic
90
10
-4
Traffic noise in a small car
80
10
-5
Vacuum cleaner
70
10
-6
Normal conversation
60
10
-7
Average home
50
10
-8
Quiet conversation
40
10
-9
Soft whisper
30
10
-10
Quiet living room
20
10
-11
Quiet recording studio
10
10
-12
Barely audible
0
Decibel levels of common noises.

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Logarithms can provide an expression of relations that are inverses of expo-
nential situations. A battery charges at a rate that depends on how close it is to
being fully charged. It charges fastest when it is most discharged. The charge C
at any instant t is modeled by the formula 
C = M (1 − e
−kt
), where M is the
maximum charge that the battery can hold, and k is a constant that depends on
the battery and charger. The formula that gives the time required to charge a bat-
tery uses the natural logarithm function ln: 
t = −
1
k
ln(1 −
C
M
).
Logarithms appear in a wide range of industrial and technological applica-
tions. The Haugh unit is a measure of egg quality that uses base-10 logarithms.
The logarithms of the sizes of two organs of an animal are related in an allomet-
ric equation. Economists use logarithmic derivatives to compare price changes 
of different items. The effective steam pressure in a cylinder is 
p =
P (1+ln(R))
R
,
where P is the initial absolute pressure, and R is the ratio of expansion. The num-
ber of turns of a rope or pulley about a large cylinder that would be needed to
keep the rope from slipping is found with a formula that uses logarithms of ten-
sion ratios. Electrical engineers use Bode plots, a form of logarithmic graphing,
to determine voltage gains for active or passive filters. Statisticians use loga-
rithms to linearize data that appear to lie in certain curvilinear patterns.
online sources for further exploration
Investigate pH 



e as the base of natural logarithms at the MathSoft site

CoolMath’s table of decibel levels at 

National earthquake information center

An excellent list of applications in technical areas is at British Columbia Institute 
of Technology

Sonic booms and logarithms

Sound pressure levels and intensity


Belt friction

Bode plots