1. The Logarithmic Function
Lesson 4.3

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

4. 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

5. Graph, Domain, Range Note domain for 0 < x < 1
Change the x to 5, view again

6. Graph, Domain, Range
View graph with window -1 < x < 10, -4 < y < 5
Why does the graph appear undefined for x < 0 ?

7. 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 }

8. Vertical Asymptote
Note behavior of function as x  0+

9. Inverse Functions
Recall use of the DrawInv command on the graph screen
You type in y1(x)

10. Inverse Functions Now consider the 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 symmetric about y = x
Shows they are inverses

11. Assignment
Lesson 4.3A
Page 173
Exercises
1 – 11 odd, 19 – 31 odd

13. 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

14. 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

15. Chemical Acidity
What would be the hydrogen ion concentration of vinegar with pH = 3?

16. 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

17. 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

18. 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
The log of the ratio

19. Decibels Approx. Decibel Level Example
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.

20. 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?

21. Measuring Earthquakes
S-wave
Surface-wave
P-wave
Pressure-wave

22. Measuring Earthquakes

23. Measuring Earthquakes
Seismic waves radiated by all earthquakes can provide good estimates of their magnitudes

24. 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

25. Comparable Magnitudes
Richter TNT for Seismic Example
Magnitude Energy Yield (approximate)
ounces Breaking a rock on a lab table
pounds Large Blast at a Construction Site
pounds
ton Large Quarry or Mine Blast
tons
tons
tons
,000 tons Small Nuclear Weapon
,100 tons Average Tornado (total energy)
,000 tons
,000 tons Little Skull Mtn., NV Quake, 1992
million tons Double Spring Flat, NV Quake, 1994
million tons Northridge, CA Quake, 1994
million tons Hyogo-Ken Nanbu, Japan Quake, 1995; Largest Thermonuclear Weapon
million tons Landers, CA Quake, 1992
billion tons San Francisco, CA Quake, 1906
billion tons Anchorage, AK Quake, 1964
billion tons Chilean Quake, 1960
trillion tons (San-Andreas type fault circling Earth)
trillion tons (Fault Earth in half through center,
OR Earth's daily receipt of solar energy)

26. Assignment
Lesson 4.3B
Page 174
Exercises
13 – 17 all, 33 – 37 all