WORKSHEET KEY 2 1/11/2017 11:53 PM c 5.5 - Properties of Logarithms 1 NATURAL LOGARITHMS Section 5.5A, Revised ©2012, [email protected] 1/11/2017 11:53 PM 5.5 - Properties of Logarithms 2 CONVERSIONS A. The base of Natural Log (ln) is e. B. Convert: lne x = y to ey = x 1/11/2017 11:53 PM 5.5 – Properties of Logarithms 3 EXAMPLE 1 Convert ln 2/5 = –0.916 to exponential form ln e x y y e =x e 1/11/2017 11:53 PM 0.916 2 5 5.5 – Properties of Logarithms 4 YOUR TURN Convert ln 679 = 6.520 to exponential form e 1/11/2017 11:53 PM 6.520 679 5.5 – Properties of Logarithms 5 EXAMPLE 2 Convert e2 = 7.389 to logarithmic form e y x ln y x ln7.3890 2 1/11/2017 11:53 PM 5.5 – Properties of Logarithms 6 YOUR TURN Convert e2x = 3 to logarithmic form ln3 2x 1/11/2017 11:53 PM 5.5 – Properties of Logarithms 7 SIMPLIFYING NATURAL LOGARITHMS A. The inverse of a natural base (e) is the natural log (ln) B. If there is ln and e, they cancel each other out C. Natural logarithms have the same properties as log base 10 and logarithms with other bases D. The base of a natural log is e but it will never be written as the base. E. ln + e = FELONY 1/11/2017 11:53 PM 5.5 – Properties of Logarithms 8 NATURAL LOGARITHM STEPS A. RAISE IT UP by incorporating e as the base to both sides B. Cancel any ln e ’s C. Simplify using Natural Logarithm rules D. Check ln e x 1/11/2017 11:53 PM e 5.5 - Properties of Logarithms ln x 9 EXAMPLE 3 Solve ln x = 4 ln x 4 ln x e 4 x 54.598 1/11/2017 11:53 PM 5.5 - Properties of Logarithms 10 EXAMPLE 4 Solve 5 + 2 ln x = 7 x 2.718 1/11/2017 11:53 PM 5.5 - Properties of Logarithms 11 YOUR TURN Solve 3 ln x – 6 = 9 x 148.413 1/11/2017 11:53 PM 5.5 - Properties of Logarithms 12 EXAMPLE 5 Solve ex + 5 = 60 e 5 60 x e 55 x ln e ln 55 x x 4.007 1/11/2017 11:53 PM 5.5 - Properties of Logarithms 13 EXAMPLE 6 Solve –14 + 3ex = 11 x 2.120 1/11/2017 11:53 PM 5.5 - Properties of Logarithms 14 YOUR TURN Solve 7 – 2ex = 5 x0 1/11/2017 11:53 PM 5.5 - Properties of Logarithms 15 REVIEW Solve x2 – 3x + 2 = 0 x 3x 2 0 2 x 2 x 1 0 1, 2 1/11/2017 11:53 PM 5.5 - Properties of Logarithms 16 EXAMPLE 7 Solve e2x – 3ex + 2 = 0 e 3e 2 0 2x x e 3e e 2 e x 2 x 20 x x 1 0 e 20 e 1 0 x 1/11/2017 11:53 PM x 5.5 - Properties of Logarithms 17 EXAMPLE 7 Solve e2x – 3ex + 2 = 0 e 20 x e 2 x ln e ln 2 x ln e ln 2 x ln 2 x 0.693 e 1 0 x e 1 x ln e ln1 x ln e ln1 x ln1 x0 x 1/11/2017 11:53 PM x 0.693,0 5.5 - Properties of Logarithms 18 EXAMPLE 8 Solve e2x – 7ex + 12 = 0 1.099, 1.386 1/11/2017 11:53 PM 5.5 - Properties of Logarithms 19 EXAMPLE 9 Solve 2e2x + 7ex – 4 = 0 0.693ext : ln 4 1/11/2017 11:53 PM 5.5 - Properties of Logarithms 20 YOUR TURN Solve 6e2x + 11ex – 2 = 0 1.792ext : ln 2 1/11/2017 11:53 PM 5.5 - Properties of Logarithms 21 EXAMPLE 10 You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double? rt Doubled Amount A Pe 0.0675t A 500e 0.0675t 1000 500e 0.0675t 500e 1000 0.0675t 500e 500 1/11/2017 11:53 PM 1000 500 5.5 - Properties of Logarithms 22 EXAMPLE 10 You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double? rt A Pe 0.0675 t e 2 0.0675t ln e ln 2 0.0675t ln 2 ln 2 t 0.0675 10.269 years 1/11/2017 11:53 PM 5.5 - Properties of Logarithms 23 EXAMPLE 11 You have deposited $2,500 in an account that pays 8.5% interest, compounded continuously. How long will it take your money to triple? 12.925years 1/11/2017 11:53 PM 5.5 - Properties of Logarithms 24 YOUR TURN How long will it take $30,000 to accumulate to $110,000 in a trust that earns a 10% annual return compounded continuously? 12.993years 1/11/2017 11:53 PM 5.5 - Properties of Logarithms 25 EXAMPLE 12 During its exponential growth phase, a certain bacterium can grow from 5,000 cells to 12,000 cells in 10 hours. What is the growth rate? kt P = 12,000 Ending Amount 0 P Pe k 10 12,000 5,000e 12,000 k 10 e 5,000 12 k 10 ln ln e 5 1/11/2017 11:53 PM P0== Initial 5,000 Amount e = The Natural Base kK = ?? Growth or Decay Rate t = 10 T = Time 5.5 - Properties of Logarithms 26 EXAMPLE 12 During its exponential growth phase, a certain bacterium can grow from 5,000 cells to 12,000 cells in 10 hours. What is the growth rate? 12 k 10 ln ln e 5 12 ln 10k ln e 5 12 ln 5 k 10 k 0.0875 1/11/2017 11:53 PM 5.5 - Properties of Logarithms 27 EXAMPLE 13 During its exponential growth phase, a certain bacterium can grow from 5,000 cells to 15,000 cells in 12 hours. What is the growth rate? 1/11/2017 11:53 PM k 0.0926 5.5 - Properties of Logarithms 28 YOUR TURN The population of a certain city in 2000 was 99,500. What is its initial population in 1975 when its growth rate is at 0.170. Round to the nearest whole number. 1/11/2017 11:53 PM P0 65,050 5.5 - Properties of Logarithms 29 ASSIGNMENT Worksheet 1/11/2017 11:53 PM 5.5 - Properties of Logarithms 30
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