7.3 LOGARITHMIC FUNCTIONS

DECIBELS

Sound is measured in units called decibels. Decibel level begins at zero, which is near total silence.

A whisper is at 30 decibels, and a normal conversation is 60 decibels.

An increase of 10 means that a sound is 10 times more intense.

So, how much more intense is a normal conversation than a whisper?

TRY THIS ONE:

An ambulance siren is at 120 decibels.

How many more times is that than the weakest sound we can hear (which is at 0)?

What is the base we are looking at?

WHY DO WE CARE?

Listening to sounds at high decibel levels for a long period of time can cause hearing loss.

Those who are exposed to noise levels at 85 decibels or higher for a prolonged period of time are at risk for hearing loss.

MP3’s on loud volumes are 105 decibels. How many more times more intense is that

than where the damage will occur?

SO WHAT?

So what does this have to do with these exponential functions we’ve been working with?

Parts of an exponential equation:

bx = a

THINK ABOUT THIS:

How many times would you have to double $1 before you had $8?

LOGARITHMS

A logarithm would help us be able to solve that type of problem, where we’re looking for the exponent.

It is the exponent to which a specified base is raised to obtain a given value.

An exponential equation and a logarithmic equation are related:

Exponential Logarithmic bx = a logba = x

(means log base b of a is x)

REWRITE EACH EXPONENTIAL AS A LOGARITHM

26

92

x0

WRITE EACH LOGARITHMIC AS AN EXPONENTIAL

log12 144 = 2

log ½ 8 = -3

log8 0.125 =?

SPECIAL PROPERTIES OF LOGS

For an base b such that b>0 and b ≠ 1:

logb b = 1

logb 1 = 0

THE MOST COMMON LOG

A logarithm with base 10 is the common logarithm.

This means, if there is no base, then you can assume the base is 10.

Example: log 5 means log10 5

This is the log that is on your calculator.

FIND THE FOLLOWING LOGS USING MENTAL MATH

log 1000 (think: 10 to what power is 1000?)

log4 ¼ (think: 4 to what power is ¼?)

log 0.00001

log250.04

SOLVE FOR THE UNKNOWN:

log8 x = 3

logx 343 = 3

log5 625 = x