1/3/2007 Pre-Calculus State “exponential growth” or “exponential decay” (no calculator needed) State “exponential growth” or “exponential decay” (no calculator needed) a.) y = e 2x b.) y = e –2x c.) y = 2 –x d.) y = 0.6 –x k > 0, exponential growth k < 0, exponential decay b>1 so growth, but reflect over y-axis, so decay 0<b<1 so decay, but reflect over y-axis, so growth

1/3/2007 Pre-Calculus Characteristics of a Basic Exponential Function: Domain: Range: Continuity: Symmetry: Boundedness: Extrema: Asymptotes: End Behavior: ( - ,  ) ( 0,  ) continuous none b = 0 none y = 0 lim f(x) x   =  lim f(x) x  -  = 0

1/3/2007 Pre-Calculus

1/3/2007 Pre-Calculus Question Use properties of logarithms to rewrite the expression as a single logarithm. log x + log y 1/5 log z log x + log 5 2 ln x + 3 ln y ln y – ln 34 log y – log z ln x – ln y4 log (xy) – 3 log (yz) 1/3 log x3 ln (x3y) + 2 ln (yz2)

1/3/2007 Pre-Calculus Change of Base Formula for Logarithms

1/3/2007 Pre-Calculus “the” exponential function the “natural base” (irrational, like  ) Leonhard Euler (1707 – 1783) f(x) = a e kx for an appropriately chosen real number, k, so e k = b exponential growth function exponential decay function

1/3/2007 Pre-Calculus State “exponential growth” or “exponential decay” (no calculator needed) State “exponential growth” or “exponential decay” (no calculator needed) a.) y = e 2x b.) y = e –2x c.) y = 2 –x d.) y = 0.6 –x k > 0, exponential growth k < 0, exponential decay b>1 so growth, but reflect over y-axis, so decay 0>b>1 so decay, but reflect over y-axis, so growth

1/3/2007 Pre-Calculus Rewrite with e; approximate k to the nearest tenth. a.) y = 2 x b.) y = 0.3 x y = e 0.7x y = e –1.2x e ? = 2e ? = 0.3

1/3/2007 Pre-Calculus Characteristics of a Basic Logistic Function: Domain: Range: Continuity: Symmetry: Boundedness: Extrema: Asymptotes: End Behavior: ( - ,  ) ( 0, 1 ) continuous about ½, but not odd or even B = 0, b = 0 none y = 0, 1 lim f(x) x   = 1 lim f(x) x  -  = 0

1/3/2007 Pre-Calculus Based on exponential growth models, will Mexico’s population surpass that of the U.S. and if so, when? Based on logistic growth models, will Mexico’s population surpass that of the U.S. and if so, when? What are the maximum sustainable populations for the two countries? Which model – exponential or logistic – is more valid in this case? Justify your choice.

1/3/2007 Pre-Calculus Logarithmic Functions inverse of the exponential function log b n = p b p = n log b n = p iff b p = n find the power 2 ? = 32 = 5 3 ? = 1 = 0 4 ? = 2 = ½ 5 ? = 5 = 1 2 ? =  2 = ½

1/3/2007 Pre-Calculus Basic Properties of Logarithms (where n > 0, b > 0 but ≠ 1, and p is any real number) Basic Properties of Logarithms (where n > 0, b > 0 but ≠ 1, and p is any real number) log b 1 = 0because b 0 = 1 log b b = 1because b 1 = b log b b p = pbecause b p = b p b log b n = nbecause log b n = log b n Example log 5 1 = 0 log 2 2 = 1 log = 3 6 log 6 11 = 11

1/3/2007 Pre-Calculus Evaluating Common Log Expressions log 100 = 10 log 8 = Without a Calculator: log 32.6 = log 0.59 = log (–4) = With a Calculator: log 7  10 = 2 1/ – … undefined

1/3/2007 Pre-Calculus Solving Simple Equations with Common Logs and Exponents Solve: 10 x = 3.7 x = log 3.7 log x = – 1.6 x ≈ 0.57 x = 10 –1.6 x ≈ 0.03

1/3/2007 Pre-Calculus Evaluating Natural Log Expressions log e 7 = e ln 5 = Without a Calculator: ln 31.3 ln 0.39 ln (–3) With a Calculator: ln 3  e = 1/3 7 5 ≈ ≈ – = undefined

1/3/2007 Pre-Calculus Solving Simple Equations with Natural Logs and Exponents Solve: ln x = 3.45 x = e 3.45 e x = 6.18 x ≈ x = ln 6.18 x ≈ 1.82

1/3/2007 Pre-Calculus Logarithmic Functions ≈ 0.91 ln x vertical shrink by 0.91 ≈ – 0.91 ln x reflect over the x-axis vertical shrink by 0.91

1/3/2007 Pre-Calculus f(x) = log 4 x f(x) = log 5 x f(x) = log 7 (x – 2) Graph the function and state its domain and range: f(x) = log 3 (2 – x) ln x ln x ln (x – 2) ln (–(x – 2) Vertical shrink by Vertical shrink by Vertical shrink by 0.514, shift right 2 Vertical shrink by Reflect across y-axis Shift right 2

1/3/2007 Pre-Calculus Logarithmic Functions one-to-one functions u = v isolate the exponential expression take the logarithm of both sides and solve 2 x = 2 5 x = 5 log 2 2 x = log 2 7 x = log 2 7

1/3/2007 Pre-Calculus Newton’s Law of Cooling An object that has been heated will cool to the temperature of the medium in which it is placed (such as the surrounding air or water). The temperature, T, of the object at time, t, can be modeled by: where T m = temp. of surrounding medium T 0 = initial temp. of the object Example: A hard-boiled egg at temp. 96  C is placed in 16  C water to cool. Four (4) minutes later the temp. of the egg is 45  C. Use Newton’s Law of Cooling to determine when the egg will be 20  C.

1/3/2007 Pre-Calculus Compound Interest Interest Compounded Annually A = P (1 + r) t A = AmountP = Principalr = Ratet = Time Interest Compounded k Times Per Year A = P (1 + r/k) kt k = Compoundings Per Year Interest Compounded Continuously A = Pe rt

1/3/2007 Pre-Calculus Annual Percentage Yield APY = (1 + r/k) k – 1 Compounded Continuously APY = e r – 1

1/3/2007 Pre-Calculus Annuities R = Value of Payments i = r/k = interest rate per compounding n = kt = number of payments Future Value of an Annuity # 11 (p. 324) $14,755.51

1/3/2007 Pre-Calculus Annuities R = Value of Payments i = r/k = interest rate per compounding n = kt = number of payments Present Value of an Annuity For loans, the bank uses a similar formula

1/3/2007 Pre-Calculus Annuities If you loan money to buy a truck for $27,500, what are the monthly pay- ments if the annual percentage rate (APR) on the loan is 3.9% for 5 years?