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Chapter 4

Chapter 4. 4-6 The Natural base, e. Quiz#1. Express each as a single logarithm. 1. Simplify. 3 4. 5. Quiz#1. Solve. 6 . 7 . 8 . lo g7 ( 5 x+ 3 )= 3 9 . log ( 3 x + 1 ) - log 4 = 2 10. lo g4 ( x- 1)+ lo g4 (3 x- 1)=2. Objectives.

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Chapter 4

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  1. Chapter 4 4-6 The Natural base, e

  2. Quiz#1 • Express each as a single logarithm. • 1. • Simplify. • 3 • 4. • 5.

  3. Quiz#1 • Solve. • 6. • 7. • 8. lo g7 ( 5x+3 )= 3 • 9. log ( 3x +1 ) - log 4 = 2 • 10. lo g4 (x-1)+ lo g4 (3x- 1)=2

  4. Objectives Use the number e to write and graph exponential functions representing real-world situations. Solve equations and problems involving e or natural logarithms.

  5. Natural Base • Recall the compound interest formulaA = P(1 + r/n )nt, where A is the amount, P is the principal, r is the annual interest, n is the number of times the interest is compounded per year and t is the time in years.

  6. Natural base • Suppose that $1 is invested at 100% interest (r = 1) compounded n times for one year as represented by the function f(n) = P(1 + 1/n )n.

  7. Natural base • As n gets very large, interest is continuouslycompounded. Examine the graph of f(n)= (1 + 1/n )n. The function has a horizontal asymptote. As n becomes infinitely large, the value of the function approaches approximately 2.7182818…. This number is called e. Like , the constant e is an irrational number.

  8. Natural base

  9. Exponential functions • Exponential functions with e as a base have the same properties as the functions you have studied. The graph of f(x) = ex is like other graphs of exponential functions, such as f(x) = 3x The domain of f(x) = ex is all real numbers. The range is {y|y > 0}.

  10. Example 1 • Graph f(x) = ex–2 + 1. • Make a table. Because e is irrational, the table values are rounded to the nearest tenth.

  11. Example 2 • Graph f(x) = ex – 3.

  12. Student guided practice • Do problems 2-5 in your book page 278

  13. Natural logs • A logarithm with a base of e is called a natural logarithmand is abbreviated as “ln” (rather than as loge). Natural logarithms have the same properties as log base 10 and logarithms with other bases.

  14. Natural logs • The natural logarithmic functionf(x) = lnx is the inverse of the natural exponential function f(x) = ex.

  15. Natural logs • The domain of f(x) = lnx is {x|x> 0}. • The range of f(x) = lnx is all real numbers • All of the properties of logarithms from Lesson 4-4 also apply to natural logarithms.

  16. Example #3 • Simplify. • A. lne0.15t • B. e3ln(x +1) • C. lne2x + lnex

  17. Student guided practice • Do problems 6-10 in your book page 278

  18. ContinuouslyCompounded • The formula for continuously compounded interest is A = Pert, where A is the total amount, P is the principal, r is the annual interest rate, and t is the time in years.

  19. Example • What is the total amount for an investment of $500 invested at 5.25% for 40 years and compounded continuously?

  20. Example • What is the total amount for an investment of $100 invested at 3.5% for 8 years and compounded continuously?

  21. Student guided practice • Do problem 11 in your book page 278

  22. Half life formula • The half-life of a substance is the time it takes for half of the substance to breakdown or convert to another substance during the process of decay. Natural decay is modeled by the function below.

  23. Example • Pluonium-239 (Pu-239) has a half-life of 24,110 years. How long does it take for a 1 g sample of Pu-239 to decay to 0.1 g?

  24. Example • Determine how long it will take for 650 mg of a sample of chromium-51 which has a half-life of about 28 days to decay to 200 mg.

  25. Student guided practice • Do problem 12 in your book page 278

  26. Homework • Do odd problems from 13 to 22 in your book page 278

  27. Closure • Today we learned about the natural base of e • Next class were are going to learn about transforming exponential and logarithmic function

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