2.06 ExpLog - Natural Logarithm

2 Exponential and Logarithmic Functions 2.06 Natural Logarithms A Special Case Logarithms with a BASE 10 are called “Common Logarithms” 10𝑦 = 𝑥 ⇒ 𝑦 = log 𝑥 Logarithms with a BASE e are called “Natural Logarithms” 𝑒 𝑦 = 𝑥 ⇒ 𝑦 = log 𝑒 𝑥 𝑦 = ln 𝑥 𝐿𝑛: 𝑙𝑜𝑔𝑎𝑟𝑖𝑡ℎ𝑚𝑒 𝑛𝑎𝑡𝑢𝑟𝑒𝑙 Natural v Common Log •Using your calculator, see 1 what 𝑒 is 𝑒 ≈ 2.71828182845904523. . Investigate • e is a value that came from the expression: 1 • 1 𝑥 + 𝑥 As 𝑥 ⟼ ∞, the value of the expression is 2.7182818284…(irrational number) • 𝑒 = lim 1 + 𝑥→∞ 1 𝑥 𝑥 → “Euler Number” • The value of e is also obtained by adding the terms in the 5! = 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 120 series: 1 1 1 1 1 𝑒 =1+ + + + + +⋯ 1! 2! 3! 4! 5! 1 1 1 1 𝑒 =1+1+ + + + +⋯ 2 6 24 120 𝑒 ≈ 2.71828182845904523. . What is “e” • In science and math, the base e is very important. • As 𝜋 is known to relate a circle’s diameter to its 𝑐 circumference, 𝜋 = , e will be used in calculus 𝑑 (derivatives). • “unique function which is its own derivative” 𝑥 𝑑𝑡 ln 𝑥 = න =1 1 𝑡 Importance of “e” • e appears in growth or decay problems (compound interest) • The Bell Curve (stats) • Some Probability • Study of Factorials Where do we see it? • Slope of the Tangent Line at point (0, 1) is equal to 1 Graph 𝑦1 = 𝑒 𝑥 • Slope of Tangent Line at point (0, 1) is ≈ 0.694 𝑦= 𝑥 2 • Slope of Tangent Line at point (0, 1) is ≈ 1.099 vs 𝑦= 𝑥 3 𝑒 𝑦 = 𝑥 ⇒ 𝑦 = log 𝑒 𝑥 = ln 𝑥 • 𝑦 = ln 1 means 𝑒 𝑦 = 1 ⇒𝑦=0 • 𝑦 = ln 𝑒 means 𝑒 𝑦 = 𝑒 ⇒𝑦=1 • 𝑦 = ln 10 means 𝑒 𝑦 = 10 ⇒ 𝑦 ≈ 2.3026 Usage of ln(x) • Basic: 𝑓 𝑥 = • Transformed: 𝑒𝑥 −1, 1 𝑒 0,1 (1, 𝑒) 𝑓 𝑥 = 𝑎𝑒 𝑏(𝑥−ℎ) + 𝑘 • Parameters: Same as previous exponential functions • Sketching: Use OPF (basic) ⟼ (transformed) • Properties: Same as transformed exponential with 𝑐 > 1 (since 𝑒 ≈ 2.72) Exponential “BASE e” Function Properties of 𝒇 𝒙 = 𝒆 Domain Range Intercepts Initial Value Zeros Variation Extremes Sign Miss Connell 𝒙 ℝ 𝑎 > 0 → ]𝑘, +∞[ 𝑎 < 0 → ] − ∞, 𝑘[ 𝑓(0) 𝑎 and 𝑘 same sign → 𝑛𝑜𝑛𝑒 𝑎 and 𝑘 opposite signs → 𝑜𝑛𝑒 𝑎 > 0 → 𝑠𝑡𝑟𝑖𝑐𝑡𝑙𝑦 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑎 < 0 → 𝑠𝑡𝑟𝑖𝑐𝑡𝑙𝑦 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑁𝑂𝑁𝐸 𝑖𝑓 𝑎 > 0 and 𝑘 < 0 → Changes at the zero 𝑖𝑓 𝑎 > 0 and 𝑘 > 0 → strictly positive 𝑖𝑓 𝑎 < 0 and 𝑘 < 0 → strictly negative 𝑖𝑓 𝑎 < 0 and 𝑘 > 0 → Changes at the zero 𝑦 1.Isolate the 𝑒 2.Switch forms to write as ln(x) 𝑦 𝑒 = 𝑥 ⇒ 𝑦 = ln 𝑥 3.Solve with Calculator using 𝑙𝑛 button Finding the Zero • 𝑓 𝑥 = 2𝑒 3𝑥−4 − 5 0 = 2𝑒 3𝑥−4 − 5 5 = 2𝑒 3𝑥−4 5 = 𝑒 3𝑥−4 2 ln 2.5 = 3𝑥 − 4 0.9163 ≈ 3𝑥 − 4 4.9163 ≈ 3𝑥 1. 6388 ≈ 𝑥 Example • Solve for Zero • Isolate 𝑒 𝑦 • Switch forms 𝑦 = ln 𝑥 ⇒ 𝑒 𝑦 = 𝑥 • Solve for x Equations: same as zeroes (isolate exponential part) Inequalities: 1) sketch 2) solve intersection point 3) express solution interval Equations & Inequalities • −3𝑒 𝑥−3 + 8 < 3 1. Solve for the equation: −3𝑒 𝑥−3 + 8 = 3 −3𝑒 𝑥−3 = −5 5 𝑥−3 𝑒 = 3 𝑥 − 3 = ln(1.6667) 𝑥 − 3 = 0.510845623 𝑥 = 3.510845623 𝑥 ∈ ]3.5108, +∞[ Inequalities – Ex 1 • Recall: • e is a value that came from the expression: 1 • 1 𝑥 + 𝑥 As 𝑥 ⟼ ∞, the value of the expression is 2.7182818284…(irrational number) • 𝑒 = lim 1 + 𝑥→∞ 1 𝑥 𝑥 → “Euler Number” • Looks like compound interest formula…. •C 𝑥 = 𝐶𝑜 • • • • 𝑟𝑥 (𝑒) C(x) = capital after “x” years Co = amount invested r = annual interest rate (decimal) x = number of years Continuously Compounding Interest • Sarah invests $10,000 in a savings account with a fixed annual interest rate of 5.50% compounded continuously. What will be the account balance after 12 years? • 𝐶 12 = 10000𝑒 0.055 12 • 𝐶 12 = 10000(1.934792334) • 𝐶 12 ≈ 19347.92 Ex 2: Financial • Compound Interest practice sheet 3, 4, 7-10 • Solving Exponential Equations with Logarithms worksheet #10, #19-26 Homework