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Section 3.5

Section 3.5. Applications of Logarithims. One-to-One Nature of Exponential and Logarithmic Functions. Examples:. Logs are used to express values that would become impractically large or small. Large distances… from earth to sun, from earth to moon, “width” of Milky Way

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Section 3.5

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  1. Section 3.5 Applications of Logarithims

  2. One-to-One Nature of Exponential and Logarithmic Functions Examples:

  3. Logs are used to express values that would become impractically large or small • Large distances… from earth to sun, from earth to moon, “width” of Milky Way • Small distances/mass… width of molecule, mass of electron. • Intensity of earthquakes – Richter Scale • Orders of magnitude – How many factors of 10 larger/smaller?

  4. Distances • Earth to Moon: 3.8 x 108 meters • Or 380 million meters • Earth to Sun: 1.47 x 1011 meters • Or 147 BILLION meters • Approximately 103 times farther • or 3 orders of magnitude • Actually “only” about 387 times farther

  5. Example of DATA collected… It’s difficult to represent the graph relating x and y because the values of y vary so widely, so the graph of y vs. x looks like this: But if you look at the graph of log y vs. x, it looks like this

  6. Solving Log Equations Write in exponential form Simplify Isolate the variable “The base… raised to the answer… equals what you’re taking the log of”

  7. Multiply both sides by 3 Multiply both sides by 2x Problem: Can’t exponentiate a sum easily… get rid of 2-x Multiply through by??? 2x, so you can keep the bases the same! Simplify Using AQUAD Solve quadratics by setting = to zero, and using quadratic formula if necessary But remember, you weren’t asked to solve for u, you must “back substitute and solve for x Now you have to get creative… doesn’t this equation resemble a quadratic? Consider SUBSTITUTING u = 2x

  8. Comparing Earthquake Intensities The Richter Scale is a logarithmic scale, since the intensity of the earthquakes varies so widely. Compare the intensities of two earthquakes, one measuring 7.2 and the other measuring 5.5. What we’re really being asked for is the ratio of severities measured (a1/a2).

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