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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. Graph, Domain, Range. Use your calculator to discover facts about the log function
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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
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 }
Vertical Asymptote • Note behavior of function as x 0+
Inverse Functions • Recall use of the DrawInv command on the graph screen You type in y1(x)
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
Assignment • Lesson 4.3A • Page 173 • Exercises • 1 – 11 odd, 19 – 31 odd
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)
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
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 The log of the ratio
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 • S-wave • Surface-wave • P-wave • Pressure-wave
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
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)
Assignment • Lesson 4.3B • Page 174 • Exercises • 13 – 17 all, 33 – 37 all