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Honors Finite Section 10.1 – Review of Exponential and Logarithmic Functions A function with a variable in an exponent is called an exponential equation. The number being raised to the power is called the base. n 1 One of the most useful bases for exponential functions is e. e lim 1 2.71828... n n Value Appreciation: For an amount P that increases at rate r each year, value after t P(1 r ) t years This formula can be used to find any quantity that increases annually by a fixed percentage. For quantities that decrease, make r negative. Example 1:A Van Gogh painting was auctioned in 1996 for $10,342,500. Assuming that it continues to appreciate in value by 10% per year, predict its value in 2016. Logarithmic Functions: Common logs – base 10 : log x y is equivalent to 10 y x. Natural logarithms – base e: ln x y is equivalent to e y x. Example 2: (a) Find log 3 10 (b) Find ln e 4 Properties of Natural Logarithms (also apply to Common Logarithms) 1. ln 1 = 0 2. ln e 1 3. ln e x x 4. eln x x 5. ln( M N ) ln M ln N 1 6. ln ln N N M 7. ln ln M ln N N 8. ln M P P ln M Finite Notes - Section 10.1 – page 2 Example 3: Simplify without a calculator. x (a) ln(eln e ) (b) f ( x) ln ln 2 ln1 2 2 ln(e 4 x ) ln x e (c) f ( x) x Example 4: Biomedical: Drug Dosage If a dosage d of a drug is administered to a patient, the amount of the drug remaining in the tissues t hours later will be f (t ) d e kt where k (the “absorption constant”) depends on the drug. For the immunosuppressant cyclosporine, the absorption constant is k 0.012. For a dose of d 400 mg, use the above formula to find the amount remaining in the tissues after: (a) 24 hours (b) 48 hours Example 5: Behavioral Science: Learning In a psychology experiment, the proportion of information from a photograph that a student could remember after t minutes was e0.18t . a. What proportion was remembered after 2 minutes? b. How long did it take for the proportion to fall to 50%? c. Graph r ( x) e0.18 x on a graphing calculator (using an appropriate window) together with a horizontal line representing 50% and verify your answer to part (b) using INTERSECT.