Unit 3 Exponential Functions

Unit 3 Exponential Functions ● Law of Exponents ○ If m, n, a, and b are real numbers, with a > 0 and b > 0, then ■ am × an = am+n ■ (ab)n = an × bn 1 1 ■ a-n = 𝑎𝑛= (𝑎)n ■ (am)n = amn ■ 1m = 1 ■ A0 = 1 ○ An exponential function is function of the form: 𝑓(𝑥) = 𝑎 𝑥 ■ A is a positive real number when (a > 0) and a ≠ 1 ● a - base ● x - exponent ● y - power ● Theorem: For an exponential function f(x) = ax, if x is any real number, then ○ 𝑓(𝑥+1) 𝑓(𝑥) = 𝑎𝑥+1 = 𝑎𝑥 𝑎(𝑥+1)−(𝑥) = 𝑎1 ● Graphing Exponential functions ○ Graph 𝑓(𝑥) = 2𝑥 ■ Domain: (-∞,∞) ■ Range: (0,∞) ■ X-intercepts: none ■ Y-intercepts: (0,1) ■ Horizontal Asymptotes: y = 0, as x⟶-∞ 1 ■ Common Points: (−1, 𝑎) (0 , 1) (1 , a) ■ Continuity: (-∞,∞) 1 𝑥 ○ Graph 𝑓(𝑥) = (2) ■ Domain: (-∞,∞) ■ Range: (0,∞) ■ X-intercepts: none ■ Y-intercepts: (0,1) ■ Horizontal Asymptotes: y = 0, as x⟶∞ 1 ■ Common Points: (−1, 𝑎) (0 , 1) (1 , a) ■ Continuity: (-∞,∞) ● Graphing Using Transformations ○ Graph 𝑦 = 2−𝑥 − 3 ■ Parent function: 2x ■ d = -3, down 3 ■ b = -1, reflect x’s (y-intercept) ■ Original points: ● (-2,¼) ● (-1,½) ● (0,1) ● (1,2) ● (2,4) ■ New Points: ● (2, -2 ¾) ● (1, -2 ¼) ● (0,-2) ● (-1,-1) ● (-2,1) ■ Horizontal Asymptote: y = -3 (right) ■ Domain: (-∞,∞) ■ Range: (-3,-∞) ● The base e ○ Graph 𝑦 = 𝑒 𝑥 ■ 1 𝑛 𝑙𝑖𝑚 (1 + ) ≈ 2.71828 𝑛→∞ 𝑛 ○ Points ■ (-2,1/e2) ■ (-1,1/e) ■ (0,1) ■ (1,e) ■ (2,e2) ● Exponential Equations ○ If am = an then m = n 1. Create like bases 2. Set exponents equal ○ Solve 3x+1 = 81 ■ 3x+1 = 34 ■ x+1 = 4 ■ x=3 Logarithmic Functions and Properties ● 𝑦 = 𝑙𝑜𝑔𝑎 𝑥 if and only if 𝑥 = 𝑎 𝑦 ○ exponent = logbasepower ○ Exponential ■ y = ax ■ a = base ■ x = exponent ■ y = power ○ Example 1: eb = 9 ■ b = loge(9) ○ Example 2: log216 = 16 ■ 2x = 16 = 24 ■ x=4 ● Domain/Range of a Logarithmic Function ○ The inverse function of a logarithmic function is exponential. ○ Domain of a logarithmic function = Range of Exponential function ■ (0,∞) ○ Range of logarithmic function = Domain of Exponential function ■ (-∞,∞) ○ Example: Find the domain of 𝐹(𝑥) = 𝑙𝑜𝑔2 (𝑥 − 2) ■ x-5 > 0 ■ x>5 ■ {x| x > 5} or (5,∞) ● Graphs of Logarithmic Functions ○ 𝑓(𝑥) = 𝑙𝑜𝑔𝑎 𝑥 ○ Domain: (0,∞) ○ Range: (-∞,∞) ○ X-intercepts: (1,0) ○ Y-intercept: None ○ Asymptote: Vertical; x=0 ○ Common points: (1/a, -1) (1,0) (a,1) ○ Continuity: (0,∞) ● Natural Logarithms ○ y = logex = lnx if and only if x = ey ● Common Logarithms ○ y = logx if and only if x = 10y ● Solving Logarithmic & Exponential Equations ● Properties of Logarithms ○ log1a = 0 ○ loga0 = undefined ○ logaa = 1 ○ 𝑙𝑛𝑒 = 1 ○ log10 =1 ○ 𝑎𝑙𝑜𝑔𝑎𝑀 = 𝑀 ■ Log in exponent.... Same base ■ Cancel out if they have same base ○ Log(Product) = sum of logs ■ loga(MN) = logaM + logaN ○ Log(Quotient) = difference of logs 𝑀 ■ 𝑙𝑜𝑔𝑎 ( 𝑁 ) = 𝑙𝑜𝑔𝑎 𝑀 − 𝑙𝑜𝑔𝑎 𝑁 ○ 𝑙𝑜𝑔𝑎 𝑀𝑟 = 𝑟𝑙𝑜𝑔𝑎 𝑀 ○ Example 1: Write as sum of logs ■ 𝑙𝑜𝑔𝑎 (𝑥√𝑥 2 + 1) 1 ● 𝑙𝑜𝑔𝑎 𝑥 + 2 𝑙𝑜𝑔𝑎 (𝑥 2 + 1) ○ If M = N, then logaM = logaN ■ And vice versa ○ Change-of-Base Formula ■ 𝑙𝑜𝑔𝑎 𝑀 = 𝑙𝑜𝑔𝑏 𝑀 𝑙𝑜𝑔𝑏 𝑎 ● b = new base ○ Example 2: log589 ■ 𝑙𝑜𝑔89 𝑙𝑜𝑔5 Compound Interest ● Simple Interest Formula: 𝐼 = 𝑃𝑟𝑡 ○ I → interest ○ P → Principle (Money lent/borrowed) ○ r → Rate (percent) ○ t → time ■ Annually: 1x year ■ Semiannually: 2x year ■ Quarterly: 4x year ■ Monthly: 12x year ■ Daily: 365x year ● Compound Interest Formula 𝑟 𝑛𝑡 ○ 𝐴 = 𝑃 ⋅ (1 + 𝑛 ) 𝑟 −𝑛𝑡 ○ 𝑃 = 𝐴 ⋅ (1 + 𝑛 ) ■ A → Amount (future) P+I ■ n → number of times compounded per year ● Continuous Compounding ○ If interest was calculated infinitely over and over during the allotted time ■ A = Pert ■ P = Ae-rt ● Effective rate of interest ○ Equivalent annual simple interest rate, which would yield the same amount after 1 year as compounding ○ Needed to compare different interest rates compounded differently Exponential Growth and Decay; Newton’s Law; Logistic Models ● Growth and Decay ○ A(t) = A0ekt ○ A0 = Initial ○ When k < 0: Exponential decay ○ When k > 0: Exponential growth ● Newton’s Law of Cooling ○ 𝑢(𝑡) = 𝑇 + (𝑢0 − 𝑇) ⋅ 𝑒 𝑘𝑡 ■ u(t) - temperature after time ■ u0 - initial temperature of object ■ T - temperature of environment ■ t - time ● Logistic Growth Model 𝑐 ○ 𝑃(𝑡) = 1+𝑎𝑒 −𝑘𝑡 ■ c - carrying capacity