The Logarithmic Function

Lesson 4.3

Why?

• What happens when you enter into your calculator

• If

we want to know about limitations on the domain and range of the log function

ln 5

log yb x y x b

Graph, Domain, Range

• Use your calculator to discover facts about the log function In the Y= screen, specify log(x) Set tables with T initial x = 0, x = 0.1

• View the tables

Graph, Domain, Range

• Note domain for 0 < x < 1

• Change the x to 5, view again

Graph, Domain, Range

• View graph with window -1 < x < 10, -4 < y < 5

• Why does thegraph appearundefinedfor x < 0 ?

Graph, Domain, Range

• Recall that

• There can be no value for y that gives x < 0

• Domain for y = log x x > 0

• Range y = { all real values }

log 10yx y x

Vertical Asymptote

• Note behavior of function as x 0+

0lim logx

x

0

lim logx

x

Inverse Functions• Consider functions

y = ln x and y = ex

• Place in Y= screen Specify zoom standard, then zoom square Note relationship of the two functions Graph y = x on same graph

• Graphs are symmetricabout y = x Shows they are

inverses

Assignment

• Lesson 4.3A

• Page 173

• Exercises 1 – 11 odd, 19 – 31 odd

Usefulness of Logarithms

• Logarithms useful in measuring quantities which vary widely Acidity (pH) of a solution Sound (decibels) Earthquakes (Richter scale)

Seismologists, Frank and Earnest

Seismologists, Frank and Earnest

Chemical Acidity

• pH defined as pH = -log[H+] where [H+] is hydrogen ion concentration measured in moles per liter

• If seawater is [H+]= 1.1*10-8

then –log(1.1*10-8) = 7.96

Chemical Acidity

• What would be the hydrogen ion concentration of vinegar with pH = 3?

Logarithms and Orders of Magnitude

• Consider increase of CDs on campus since 1990 Suppose there were 1000 on campus in 1990 Now there are 100,000 on campus The log of the ratio is the change in the order of

magnitude

100,000log 2

1000

Logarithms and Orders of Magnitude

• We use the log function because it “counts” the number of powers of 10

• This is necessary because of the vast range of some physical quantities we must measure Sound intensity Earthquake intensity

Decibels

• Suppose I0 is the softest sound the human ear can hear measured in watts/cm2

• And I is the watts/cm2 of a given sound

• Then the decibels of the sound is

0

10 logI

I

The log of the

ratio

The log of the ratio

Decibels

Approx. Decibel

Level Example

0 Faintest sound heard by human ear.

30 Whisper, quiet library.

60 Normal conversation, sewing machine, typewriter.

90 Lawnmower, shop tools, truck traffic; 8 hours per day is the maximum exposure to protect 90% of people.

100 Chainsaw, pneumatic drill, snowmobile; 2 hours per day is the maximum exposure without protection.

115 Sandblasting, loud rock concert, auto horn; 15 minutes per day is the maximum exposure without protection.

140 Gun muzzle blast, jet engine; noise causes pain and even brief exposure injures unprotected ears. Maximum allowed noise with hearing protectors.

Decibels

• If a sound doubles, how many units does its decibel rating increase?

• Find out about hearing protection … How many decibels does it reduce the sound How much does that decrease the intensity of

the sound?

Measuring Earthquakes

• Seismic waves radiated by all earthquakes can provide good estimates of their magnitudes

Definition of Richter Scale

• Magnitude of an earthquake with seismic waves of size W defined as

• We measure a given earthquake relative to the strength of a "standard" earthquake

0

logW

WW

Comparable MagnitudesRichter TNT for Seismic ExampleMagnitude Energy Yield (approximate)

• -1.5 6 ounces Breaking a rock on a lab table• 1.0 30 pounds Large Blast at a Construction Site• 1.5 320 pounds• 2.0 1 ton Large Quarry or Mine Blast• 2.5 4.6 tons• 3.0 29 tons• 3.5 73 tons • 4.0 1,000 tons Small Nuclear Weapon• 4.5 5,100 tons Average Tornado (total energy)• 5.0 32,000 tons• 5.5 80,000 tons Little Skull Mtn., NV Quake, 1992• 6.0 1 million tons Double Spring Flat, NV Quake, 1994• 6.5 5 million tons Northridge, CA Quake, 1994• 7.0 32 million tons Hyogo-Ken Nanbu, Japan Quake, 1995;

Largest Thermonuclear Weapon• 7.5 160 million tons Landers, CA Quake, 1992• 8.0 1 billion tons San Francisco, CA Quake, 1906• 8.5 5 billion tons Anchorage, AK Quake, 1964• 9.0 32 billion tons Chilean Quake, 1960• 10.0 1 trillion tons (San-Andreas type fault circling Earth)• 12.0 160 trillion tons (Fault Earth in half through center, OR Earth's daily receipt of solar energy)

Assignment

• Lesson 4.3B

• Page 174

• Exercises 13 – 17 all, 33 – 37 all