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State “exponential growth” or “exponential decay” (no calculator needed) a.) y = e2x k > 0, exponential growth c.) y = 2–x b>1 so growth, but reflect over y-axis, so decay Pre-Calculus b.) y = e–2x k < 0, exponential decay d.) y = 0.6–x 0<b<1 so decay, but reflect over y-axis, so growth 1/3/2007 Characteristics of a Basic Exponential Function: Domain: ( - , ) Range: ( 0, ) Continuity: continuous Symmetry: none Boundedness: b = 0 Asymptotes: Extrema: End Behavior: lim f(x) = x lim f(x) = x- 0 Pre-Calculus none y=0 1/3/2007 Pre-Calculus 1/3/2007 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 3 4 log y – log z ln x – ln y 4 log (xy) – 3 log (yz) 1/3 log x 3 ln (x3y) + 2 ln (yz2) Pre-Calculus 1/3/2007 Change of Base Formula for Logarithms loga x log x ln x logb x or loga b log b ln b Pre-Calculus 1/3/2007 “the” exponential function the “natural base” 2.718281828459 (irrational, like ) Leonhard Euler (1707 – 1783) f(x) = a • e kx for an appropriately chosen real number, k, so ek = b exponential growth function exponential decay function Pre-Calculus 1/3/2007 State “exponential growth” or “exponential decay” (no calculator needed) a.) y = e2x k > 0, exponential growth c.) y = 2–x b>1 so growth, but reflect over y-axis, so decay Pre-Calculus b.) y = e–2x k < 0, exponential decay d.) y = 0.6–x 0>b>1 so decay, but reflect over y-axis, so growth 1/3/2007 Rewrite with e; approximate k to the nearest tenth. a.) y = 2x e? = 2 y = e0.7x Pre-Calculus b.) y = 0.3x e? = 0.3 y = e–1.2x 1/3/2007 Characteristics of a Basic Logistic Function: Domain: ( - , ) Range: ( 0, 1 ) Continuity: continuous Symmetry: about ½, but not odd or even Boundedness: B = 0, b = 0 Asymptotes: y = 0, 1 Extrema: End Behavior: lim f(x) = x- 0 lim f(x) = x 1 Pre-Calculus none 1/3/2007 Based on exponential logistic growth growth models, will Mexico’s will What Whichare model the maximum – exponential sustainable ormodels, logistic populations – isMexico’s more valid for the in population surpass of the U.S. and if so, when? this two case? countries? Justify yourthat choice. Pre-Calculus 1/3/2007 Logarithmic Functions inverse of the exponential function logbn = p bp = n logbn = p iff bp = n find the power =5 2? = 32 =0 3? = 1 =½ 4? = 2 =1 5? = 5 =½ Pre-Calculus 2? = 2 1/3/2007 Basic Properties of Logarithms (where n > 0, b > 0 but ≠ 1, and p is any real number) Example logb1 = 0 because b0 = 1 log51 = 0 logbb = 1 because b1 = b log22 = 1 logbbp = p because bp = bp log443 = 3 blogbn = n because logbn = logbn 6log611 = 11 Pre-Calculus 1/3/2007 Evaluating Common Log Expressions Without a Calculator: Pre-Calculus With a Calculator: log 100 = 2 log 32.6 = 1.5132176 log 710 = 1/7 log 0.59 = –0.22914… 10 log 8 = 8 log (–4) = undefined 1/3/2007 Solving Simple Equations with Common Logs and Exponents Solve: 10 x = 3.7 Pre-Calculus log x = – 1.6 x = log 3.7 x = 10 –1.6 x ≈ 0.57 x ≈ 0.03 1/3/2007 Evaluating Natural Log Expressions Without a Calculator: With a Calculator: ln 3e = 1/3 ln 31.3 ≈ 3.443 log e7 = 7 ln 0.39 ≈ – 0.9416 e ln 5 = 5 Pre-Calculus ln (–3) = undefined 1/3/2007 Solving Simple Equations with Natural Logs and Exponents Solve: ln x = 3.45 Pre-Calculus ex = 6.18 x = e 3.45 x = ln 6.18 x ≈ 31.50 x ≈ 1.82 1/3/2007 Logarithmic Functions ln x 1 ln x ln3 ln3 ≈ 0.91 ln x •vertical shrink by 0.91 ln x 1 ln x ≈ – 0.91 ln x ln(1/ 3) ln(1/ 3) •reflect over the x-axis •vertical shrink by 0.91 log1/ b x logb x Pre-Calculus 1/3/2007 Graph the function and state its domain and range: f(x) = log4x 0.721 ln x Vertical shrink by 0.721 f(x) = log5x 0.621 ln x Vertical shrink by 0.621 f(x) = log7(x – 2) 0.514 ln (x – 2) Vertical shrink by 0.514, shift right 2 f(x) = log3(2 – x) Pre-Calculus 0.091 ln (–(x – 2) Vertical shrink by 0.091 Reflect across y-axis Shift right 2 1/3/2007 Logarithmic Functions one-to-one functions u=v 2x = 2 5 x=5 log22x = log27 x = log27 ln7 2.81 ln 2 isolate the exponential expression take the logarithm of both sides and solve Pre-Calculus 1/3/2007 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: T(t) Tm T0 Tm ekt where Tm = temp. of surrounding medium T0 = 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. Pre-Calculus 1/3/2007 Compound Interest Interest Compounded Annually A = P (1 + r)t A = Amount P = Principal r = Rate t = Time Interest Compounded k Times Per Year A = P (1 + r/k)kt k = Compoundings Per Year Interest Compounded Continuously A = Pert Pre-Calculus 1/3/2007 Annual Percentage Yield Annual Percentage Yield APY = (1 + r/k)k – 1 Compounded Continuously APY = er – 1 Pre-Calculus 1/3/2007 Annuities Future Value of an Annuity (1 i) 1 FV R i n R = Value of Payments i = r/k = interest rate per compounding n = kt = number of payments # 11 (p. 324) Pre-Calculus $14,755.51 1/3/2007 Annuities For loans, the bank uses a similar formula Present Value of an Annuity 1 (1 i) PV R i n R = Value of Payments i = r/k = interest rate per compounding n = kt = number of payments Pre-Calculus 1/3/2007 Annuities 1 (1 i) PV R i n If you loan money to buy a truck for $27,500, what are the monthly payments if the annual percentage rate (APR) on the loan is 3.9% for 5 years? 0.039 12(5) 1 1 12 27500 R 0.039 12 Pre-Calculus 1 0.8231 27500 R 0.00325 R $505.29 1/3/2007