Day 5 in MAT229

Today:

• Announcements:
  • Hope that you had a nice long weekend.
  • Your first assignment is due today (and you have another problem set due tomorrow). Remember that assignments are due a week after they're assigned (by default, unless I tell you differently on the assignment page).
  • No "quiz opportunity" tomorrow -- but there's one today!

• quiz opportunity

• Review of Section 6.2:
  

  

  

  
  
  
  
  • A little more about Exponential Functions
    • Examples:
      1. #13, p. 401 (review of translations, reflections, etc., using )
      2. #17
      3. #36

    • Let's complete that lab we started last time: Exponential/Log Lab

• Now on to Section 6.3: Logarithmic Functions
  • Let's start by thinking about some real-life problems that would require the use of the inverse of an exponential function.
    • From science?
      • "How much uranium-235 will I have after one hundred years, if I start with a kilogram?"
      • "What's the difference in energy between an earthquake of 8 and an earthquake of 9 on the Richter scale?"
    • From business? "How long will it take my money to double at 5 percent interest, compounded continuously?"

  • Logs are, by definition, the inverses of the exponential functions:

    

  • The laws of logarithms are reflections of the properties of exponentials:

    

  • The derivation of the derivative of the natural log function is easy, given our work on inverses:

    

  • Another way to define the log function is as the anti-derivative of the "weird power" :

    

    (How does symmetry help us to understand the absolute value in this result?)

  • Examples:
    • #7, p. 408
    • 10
    • 26
    • 44