Exponential Models and Logarithms: Practice Exercises

Here are various Exercises to accompany the section Exponential Models and Logarithms. 

Exponential Models and Exponential Equations

  1. Write down an exponential equation in the indicated unknown for each of the following situations.  You do not need to solve your equation at this point.
    1. In 1997 the debt owed by the U.S. Federal Government was approximately $5.4 trillion.  Assuming that no effort is made to pay down the debt (and that the government does not incur additional debt, through deficit spending), and that the government must pay 4% interest compounded continuously, write an equation in t = the years since 1997, which shows when will the debt reach $10 trillion.
    2. You would like to eventually buy a condominium that originally requires a $15,000 down payment, but you only have $5000.  So you invest the $5,000 in a mutual fund that has an average return of 9% per year.  If the price of your condominium (and therefore the required down payment) is going up at 4% per year, write an equation in t = the years since you invested, which shows when you be able to afford the down payment, i.e., when the value of your investment equals the required down payment.
    3. Radioactive potassium, K40, has an approximate half-life of 2.5 million years. We know that a certain type of bone would originally contain 25 mg of K40, and it now only has contains 5 mg of K40.  Write an exponential equation in t = the time since the animal died (in millions of years) which shows how old the bone is.
    4. You are given 50 mg. of an experimental drug, and after 3 hours, a blood test shows that there is only 20 mg. left in your bloodstream.  Write an exponential equation in h = the biological half-life of the drug (in hours).

    Solution.

  2. Solve each of the equations from the previous Exercise algebraically, and simplify the result to obtain approximate solutions.  Make sure to verify your answer, either by graphing or substitution.

    Solution.

Exponential vs. Linear Models

  1. Use the theorem in the text to determine whether the following data represent exponential or power functions by looking at the corresponding semi-log and log-log plots in XFunctions.
    1. x log(x) y = f(x) log f(x)
      10
      60
      110
      160
      210      		 1.
      1.77815
      2.04139
      2.20412
      2.32222      		 9
      152
      371
      637
      977      		 0.954243
      2.18184
      2.56937
      2.80414
      2.98989

    2. x log(x) y = g(x) log g(x)
      10
      60
      110
      160
      210      		 1.
      1.77815
      2.04139
      2.20412
      2.32222      		 93
      153
      226
      316
      525      		 1.96848
      2.18469
      2.35411
      2.49969
      2.72016

    Solution.

  2. Make a rough a rough estimate of the slope in the following plots and the corresponding growth factor or exponent.
    1. If the following graph is a semi-log plot of f(x) = Cbx

      sketch in a straight line that "fits" the data, estimate its slope, and use the theorem in the text to determine the corresponding growth factor, b, of the exponential function, f.

    2. If the following graph is a log-log plot of g(x) = Cxp:

      sketch in a straight line that "fits" the data, estimate its slope, and use the theorem in the text to determine the corresponding exponent, p, of the power function, g.

    Solution.

Go to Trigonometric Functions.

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