Intermediate Algebra (11th edition)

The loudness of sound is measured in a unit

called a decibel,abbreviated dB.To measure with

this unit, we first assign an intensity of to a very

faint sound, called the threshold sound.If a par-

ticular sound has intensity I, then the decibel level

of this louder sound is

The table gives average decibel levels for some

common sounds. Any sound over 85 dB exceeds

what hearing experts consider safe. Permanent

hearing damage can be suffered at levels above

150 dB.

D10 log a

I

I 0

b.

I 0

606 CHAPTER 10 Inverse, Exponential, and Logarithmic Functions

Source:Deafness Research Foundation.

Decibel Level Example

60 Normal

conversation

90 Rush hour traffic,

lawn mower

100 Garbage truck,

chain saw,

pneumatic drill

120 Rock concert,

thunderclap

140 Gunshot blast,

jet engine

180 Rocket launching

pad

Measuring the Loudness of Sound

If music delivered through the earphones of an iPod has intensity Iof

find the average decibel level.

Substitute the given

value for I.

Use a calculator.

Round to the nearest

unit.

DL 95

D=10 log 1 3.162* 1092

D=10 log a

3.162* 109 I 0

I 0

b

D=10 log a

I

I 0

b

3.162* 109 I 0 ,

EXAMPLE 4

NOW TRY

OBJECTIVE 3 Evaluate natural logarithms using a calculator. Logarithms

used in applications are often natural logarithms,which have as base the number e.

The number e, like , is a universal constant.The letter ewas chosen to honor

Leonhard Euler, who published extensive results on the number in 1748. Since it is

an irrational number, its decimal expansion never terminates and never repeats.

p

Substitute the

given value for I.

e

eL2.718281828

A calculator with an key can approximate powers of e.

Powers of e

Logarithms with base eare called natural logarithms because they occur in natu-

ral situations that involve growth or decay. The base elogarithm of xis written ln x

(read “el en x”). The graph of y= ln xis given in FIGURE 15 on the next page.

e^2 L7.389056099, e^3 L20.08553692, e0.6L1.8221188

ex

NOW TRY
EXERCISE 4
Find the decibel level to the
nearest whole number of the
sound from a jet engine with
intensity Iof

6.312* 1013 I 0.

NOW TRY ANSWER
4.138 dB