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10.5: Base e and Natural Logarithms

10.5: Base e and Natural Logarithms. Definition of e & graph Evaluating e Definition of ln & graph Evaluating natural logs Equations with e and ln Compounding interest Inequalities. Definition of “e”. Suppose I look at the following expression: (1 + (1/x)) x

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10.5: Base e and Natural Logarithms

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  1. 10.5: Base e and Natural Logarithms Definition of e & graph Evaluating e Definition of ln & graph Evaluating natural logs Equations with e and ln Compounding interest Inequalities

  2. Definition of “e” • Suppose I look at the following expression: (1 + (1/x))x • On the calc, we can use the table feature to investigate what happens for large values of x • For large x, the expression seems to be approaching a value a bit larger than 2.7… we call this value “e”, the natural base • As x →∞, (1 + (1/x))x → e • “e” is an irrational number, like pi • “e” is often used in word problems involving growth or decay that is “continuous” • The graph of f(x) = ex represents exponential growth and the y-intercept is at (0, 1) (recall e is approximately 2.71828

  3. Use a calculator to evaluate to four decimal places. Keystrokes: 2nd [ex] 0.5 ENTER 1.648721271 Example 5-1a Answer: about 1.6487

  4. Use a calculator to evaluate to four decimal places. Keystrokes: 2nd [ex] –8 ENTER .0003354626 Example 5-1b Answer: about 0.0003

  5. Use a calculator to evaluate each expression to four decimal places. a. b. Example 5-1c Answer:1.3499 Answer:0.1353

  6. The natural logarithm • Recall from the last section that your calculator can easily evaluate common logarithms (logs with base 10) • Your calculator can also evaluate logarithms with a base of e (ex. Loge30) • The log with base e is called the natural logarithm, and is written ln (LN) • F(x) = ln x is the inverse of y = ex • F(x) = ln x resembles a typical logarithmic graph; the y-axis is an asymptote, the x-intercept is at (1,))

  7. Keystrokes: LN 3 ENTER 1.098612289 Example 5-2d Use a calculator to evaluate In 3 to four decimal places. Answer: about 1.0986

  8. Use a calculator to evaluate In to four decimal places. Keystrokes: LN 1 ÷ 4 ENTER –1.386294361 Example 5-2e Answer: about –1.3863

  9. Use a calculator to evaluate each expression to four decimal places. a. In 2 b. In Example 5-2f Answer:0.6931 Answer:–0.6931

  10. Write an equivalent logarithmic equation for . Answer: Example 5-3a

  11. Write an equivalent exponential equation for Answer: Example 5-3b

  12. Writeanequivalentexponentialorlogarithmicequation. a. b. Answer: Answer: Example 5-3c

  13. Evaluate Answer: Example 5-4a

  14. Evaluate . Answer: Example 5-4b

  15. Evaluate each expression. a. b. Answer: Example 5-4c Answer: 7

  16. Solving equations • Similar to what we’ve done in 10.2 – 10.4, BUT if you are taking a log of each side, use LN rather than the common log to save yourself one step (you can use the common log as well.. Just takes 1 more step)

  17. Solve Original equation Subtract 4 from each side. Divide each side by 3. Property of Equality for Logarithms Inverse Property of Exponents and Logarithms Divide each side by –2. Use a calculator. Example 5-5a Answer:The solution is about –0.3466.

  18. Check You can check this value by substituting –0.3466 into the original equation or byfinding the intersection of thegraphs of and Example 5-5b

  19. Solve Example 5-5c Answer: 0.8047

  20. Interest • Recall that earlier we saw an example involving interest that was compounded periodically (e.g., monthly, daily, etc. • A(t) = P(1 + (r/n))nt • Find the balance after 6 years if you deposit $1800 in an account paying 3% interest that is compounded monthly • A(6) = 1800(1 + (.03/12))12*6 • A(6) = $2154.51

  21. More on interest • What about if the interest is compounded not monthly,daily, or even every second, but CONSTANTLY? • We call this continuous compounding.. At ANY time you can instantly calculate your new balance • The formula we use for continuously compounding interest is: • A(t) = Pert • This expression stems from the fact that: • As x →∞, (1 + (1/x))x → e

  22. Continuous compounding formula Replace P with 700, r with 0.06, and t with 8. Simplify. Use a calculator. Example 5-6a Savings Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously. What is the balance after 8 years? Answer: The balance after 8 years would be $1131.25.

  23. A  2000 Replace Awith 700e(0.06)t. Write an inequality. Divide each side by 700. Property of Inequality for Logarithms Inverse Property of Exponents and Logarithms Example 5-6b How long will it take for the balance in your account to reach at least $2000? The balance is at least $2000.

  24. Divide each side by 0.06. Use a calculator. Example 5-6c Answer: It will take at least 17.5 years for the balance to reach $2000.

  25. Example 5-6d Savings Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously. a. What is the balance after 7 years? b. How long will it take for the balance in your account to reach at least $2500? Answer:$1065.37 Answer:at least 21.22 years

  26. Solve Original equation Write eachside using exponents and base e. Inverse Property of Exponents and Logarithms Divide each side by 3. Use a calculator. Example 5-7a Answer:The solution is 0.5496. Check this solution using substitutionor graphing.

  27. Inequalities • Again, similar to what we saw in 10.1 – 10.3 • Remember that for a log inequality, the expression you are taking the log OF must be positive • Ex. Ln (x + 3) < 4 • X must be greater than -3

  28. Solve Original inequality Write eachside using exponents and base e. Inverse Property of Exponents and Logarithms Add 3 to each. Divide each side by 2. Use a calculator. Example 5-7b

  29. Example 5-7c Answer:The solution is all numbersless than 7.5912 and greater than 1.5. Check this solution usingsubstitution.

  30. Solve each equation or inequality. a. b. Answer: Example 5-7d Answer: about 1.0069

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