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This interactive lesson covers equivalent expressions, simplifying with natural base e and natural log, solving equations, and continuously compounded interest. Practice evaluating logarithms, solving equations, and using calculators. Explore common logarithms and exponential functions.

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 7–6) CCSS Then/Now New Vocabulary Key Concept: Natural Base Functions Example 1: Write Equivalent Expressions Example 2: Write Equivalent Expressions Example 3: Simplify Expressions with e and the Natural Log Example 4: Solve Base e Equations Example 5: Solve Natural Log Equations and Inequalities Key Concept: Continuously Compounded Interest Example 6: Real-World Example: Solve Base e Inequalities Lesson Menu

  3. Use a calculator to evaluate log 3.4 to the nearest ten-thousandth. A. 0.5315 B. 1.5314 C. 1.2238 D. 29.9641 5-Minute Check 1

  4. Use a calculator to evaluate log 3.4 to the nearest ten-thousandth. A. 0.5315 B. 1.5314 C. 1.2238 D. 29.9641 5-Minute Check 1

  5. Solve 2x – 4 = 14. Round to the nearest ten-thousandth. A. 6.7256 B. 7.0164 C. 7.8074 D. 9.2381 5-Minute Check 2

  6. Solve 2x – 4 = 14. Round to the nearest ten-thousandth. A. 6.7256 B. 7.0164 C. 7.8074 D. 9.2381 5-Minute Check 2

  7. Solve 42p – 1 > 11p. Round to the nearest ten-thousandth. A. {p | p = 4} B. {p | p > 3.6998} C. {p | p < 3.4679} D. {p | p > 2.5713} 5-Minute Check 3

  8. Solve 42p – 1 > 11p. Round to the nearest ten-thousandth. A. {p | p = 4} B. {p | p > 3.6998} C. {p | p < 3.4679} D. {p | p > 2.5713} 5-Minute Check 3

  9. Express log4 (2.2)3 in terms of common logarithms. Then approximate its value to four decimal places. A. –3.4829 B. 1.5 C. 1.6845 D. 1.7063 5-Minute Check 4

  10. Express log4 (2.2)3 in terms of common logarithms. Then approximate its value to four decimal places. A. –3.4829 B. 1.5 C. 1.6845 D. 1.7063 5-Minute Check 4

  11. A. B. C. D.x = 2 log 5 Solve for x: 92x = 45. 5-Minute Check 5

  12. A. B. C. D.x = 2 log 5 Solve for x: 92x = 45. 5-Minute Check 5

  13. Content Standards A.SSE.2 Use the structure of an expression to identify ways to rewrite it. Mathematical Practices 7 Look for and make use of structure. CCSS

  14. You worked with common logarithms. • Evaluate expressions involving the natural base and natural logarithm. • Solve exponential equations and inequalities using natural logarithms. Then/Now

  15. natural base, e • natural base exponential function • natural logarithm Vocabulary

  16. Concept

  17. Write Equivalent Expressions A.Write an equivalent logarithmic equation for ex = 23. ex = 23 → loge 23 = x ln23 = x Answer: Example 1

  18. Write Equivalent Expressions A.Write an equivalent logarithmic equation for ex = 23. ex = 23 → loge 23 = x ln23 = x Answer: ln 23 = x Example 1

  19. Write Equivalent Expressions B.Write an equivalent logarithmic equation for e4 = x. e4 = x→ logex = 4 ln x = 4 Answer: Example 1

  20. Write Equivalent Expressions B.Write an equivalent logarithmic equation for e4 = x. e4 = x→ logex = 4 ln x = 4 Answer: ln x = 4 Example 1

  21. A. What is ex = 15 in logarithmic form? A. ln e = 15 B. ln 15 = e C. ln x = 15 D. ln 15 = x Example 1

  22. A. What is ex = 15 in logarithmic form? A. ln e = 15 B. ln 15 = e C. ln x = 15 D. ln 15 = x Example 1

  23. B. What is e4 = x in logarithmic form? A. ln e = 4 B. ln x = 4 C. ln x = e D. ln 4 = x Example 1

  24. B. What is e4 = x in logarithmic form? A. ln e = 4 B. ln x = 4 C. ln x = e D. ln 4 = x Example 1

  25. Write Equivalent Expressions A.Write ln x ≈ 1.2528 in exponential form. ln x ≈ 1.2528 → logex= 1.2528 x≈ e1.2528 Answer: Example 2

  26. Write Equivalent Expressions A.Write ln x ≈ 1.2528 in exponential form. ln x ≈ 1.2528 → logex= 1.2528 x≈ e1.2528 Answer:x ≈ e1.2528 Example 2

  27. Write Equivalent Expressions B.Write ln 25 = x in exponential form. ln 25= x → loge 25 = x 25 = ex Answer: Example 2

  28. Write Equivalent Expressions B.Write ln 25 = x in exponential form. ln 25= x → loge 25 = x 25 = ex Answer: 25= ex Example 2

  29. A. Write ln x ≈ 1.5763 in exponential form. A.x≈ 1.5763e B.x≈ e1.5763 C.e≈ x1.5763 D.e≈ 1.5763x Example 2

  30. A. Write ln x ≈ 1.5763 in exponential form. A.x≈ 1.5763e B.x≈ e1.5763 C.e≈ x1.5763 D.e≈ 1.5763x Example 2

  31. B. Write ln 47 = x in exponential form. A. 47 = ex B.e = 47x C.x = 47e D. 47 = xe Example 2

  32. B. Write ln 47 = x in exponential form. A. 47 = ex B.e = 47x C.x = 47e D. 47 = xe Example 2

  33. Simplify Expressions with e and the Natural Log A.Write 4 ln 3 + ln 6 as a single algorithm. 4 ln 3 + ln 6 = ln 34 + ln 6 Power Property of Logarithms = ln (34● 6) Product Property of Logarithms = ln 486 Simplify. Answer: Example 3

  34. Simplify Expressions with e and the Natural Log A.Write 4 ln 3 + ln 6 as a single algorithm. 4 ln 3 + ln 6 = ln 34 + ln 6 Power Property of Logarithms = ln (34● 6) Product Property of Logarithms = ln 486 Simplify. Answer: ln 486 Example 3

  35. Keystrokes: LN ) + LN ) ENTER 4 3 6 Keystrokes: LN ) ENTER 486 6.1862  Simplify Expressions with e and the Natural Log Check Use a calculator to verify the solution. Example 3

  36. Simplify Expressions with e and the Natural Log B.Write 2 ln 3 + ln 4 + ln y as a single algorithm. 2 ln 3 + ln 4 + ln y = ln 32 + ln 4 + ln y Power Property of Logarithms = ln (32● 4 ● y) Product Property of Logarithms = ln 36y Simplify. Answer: Example 3

  37. Simplify Expressions with e and the Natural Log B.Write 2 ln 3 + ln 4 + ln y as a single algorithm. 2 ln 3 + ln 4 + ln y = ln 32 + ln 4 + ln y Power Property of Logarithms = ln (32● 4 ● y) Product Property of Logarithms = ln 36y Simplify. Answer: ln 36y Example 3

  38. A. Write 4 ln 2 + In 3 as a single logarithm. A. ln 6 B. ln 24 C. ln 32 D. ln 48 Example 3

  39. A. Write 4 ln 2 + In 3 as a single logarithm. A. ln 6 B. ln 24 C. ln 32 D. ln 48 Example 3

  40. B. Write 3 ln 3 + ln + ln x as a single logarithm. 1 __ 3 A. ln 3x B. ln 9x C. ln 18x D. ln 27x Example 3

  41. B. Write 3 ln 3 + ln + ln x as a single logarithm. 1 __ 3 A. ln 3x B. ln 9x C. ln 18x D. ln 27x Example 3

  42. Divide each side by –2. Solve Base e Equations Solve 3e–2x + 4 = 10. Round to the nearest ten-thousandth. 3e–2x + 4 = 10 Original equation 3e–2x = 6 Subtract 4 from each side. e–2x = 2 Divide each side by 3. ln e–2x = ln 2 Property of Equality for Logarithms –2x = ln 2 Inverse Property of Exponents and Logarithms Example 4

  43. Solve Base e Equations x≈ –0.3466 Use a calculator. Answer: Example 4

  44. Solve Base e Equations x≈ –0.3466 Use a calculator. Answer: The solution is about –0.3466. Example 4

  45. What is the solution to the equation 2e–2x + 5 = 15? A. –0.8047 B. –0.6931 C. 0.6931 D. 0.8047 Example 4

  46. What is the solution to the equation 2e–2x + 5 = 15? A. –0.8047 B. –0.6931 C. 0.6931 D. 0.8047 Example 4

  47. Solve Natural Log Equations and Inequalities A.Solve 2 ln 5x = 6. Round to the nearest ten-thousandth. 2 ln 5x = 6 Original equation ln 5x= 3 Divide each side by 2. eln 5x = e3 Property of Equality for Exponential Functions 5x = e3eln x = x Divide each side by 5. x≈ 4.0171 Use a calculator. Answer: Example 5

  48. Solve Natural Log Equations and Inequalities A.Solve 2 ln 5x = 6. Round to the nearest ten-thousandth. 2 ln 5x = 6 Original equation ln 5x= 3 Divide each side by 2. eln 5x = e3 Property of Equality for Exponential Functions 5x = e3eln x = x Divide each side by 5. x≈ 4.0171 Use a calculator. Answer: about 4.0171 Example 5

  49. Solve Natural Log Equations and Inequalities B.Solve the inequality ln (3x + 1)2 > 8. Round to the nearest ten-thousandth. ln (3x + 1)2 > 8 Original equation eln (3x + 1)2> e8 Write each side using exponents and base e. (3x + 1)2 > (e4)2eln x = x and Power of of Power 3x + 1 > e4 Property of Inequality for Exponential Functions 3x > e4 – 1 Subtract 1 from each side. Example 5

  50. Solve Natural Log Equations and Inequalities Divide each side by 3. x > 17.8661 Use a calculator. Answer: Example 5

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