Understanding Logarithmic Functions: Graphs, Domains, and Ranges

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 thegraph appearundefinedfor 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 functionsy = 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

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

13. Seismologists, Frank and Earnest Usefulness of Logarithms • Logarithms useful in measuring quantities which vary widely • Acidity (pH) of a solution • Sound (decibels) • Earthquakes (Richter scale)

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

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) • -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)

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