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Understanding the Natural Logarithm and Base e in Calculus

This resource provides an in-depth exploration of the natural logarithm (ln) and the mathematical constant e, approximately equal to 2.71828. Begin with the definition and graph of e, followed by the evaluation of natural logarithms. Learn about their applications in continuous growth and decay scenarios, such as compounding interest. Key examples illustrate solving equations and inequalities involving e and ln. Gain insights on how the natural logarithm serves as the inverse of exponential functions, enhancing your understanding of calculus concepts.

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Understanding the Natural Logarithm and Base e in Calculus

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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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