1. Splash Screen

2. Concept

3. Example 1 Write Equivalent Expressions A. Write an equivalent logarithmic equation for e x = 23. e x = 23 → log e 23= x ln 23= x Answer: ln 23 = x

4. Example 1 Write Equivalent Expressions B. Write an equivalent logarithmic equation for e 4 = x. e 4 = x → log e x= 4 ln x= 4 Answer: ln x = 4

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

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

7. Example 2 Write Equivalent Expressions A. Write ln x ≈ 1.2528 in exponential form. ln x ≈ 1.2528 → log e x = 1.2528 x≈ e 1.2528 Answer: x ≈ e 1.2528

8. Example 2 Write Equivalent Expressions B. Write ln 25 = x in exponential form. ln 25 = x → log e 25 = x 25= e x Answer: 25 = e x

9. Example 2 A.x ≈ 1.5763 e B.x ≈ e 1.5763 C.e ≈ x 1.5763 D.e ≈ 1.5763 x A. Write ln x ≈ 1.5763 in exponential form.

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

11. Example 3 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 3 4 + ln 6Power Property of Logarithms = ln (3 4 ● 6)Product Property of Logarithms = ln 486Simplify. Answer: ln 486

12. Example 3 Simplify Expressions with e and the Natural Log CheckUse a calculator to verify the solution. 4 3 6 LNENTER)+LN Keystrokes: ) 486 6.1862  LNENTER) Keystrokes:

13. Example 3 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 3 2 + ln 4 + ln yPower Property of Logarithms = ln (3 2 ● 4 ● y)Product Property of Logarithms = ln 36ySimplify. Answer: ln 36y

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

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

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

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

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

19. Example 5 Solve Natural Log Equations and Inequalities A. Solve 2 ln 5x = 6. Round to the nearest ten-thousandth. Answer: about 4.0171 2 ln 5x= 6Original equation ln 5x= 3Divide each side by 2. 5x= e 3 e ln x = x Divide each side by 5. x≈ 4.0171Use a calculator.

20. Example 5 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 > 8Original equation (3x+1)²> e 8 Write in exponential form (3x + 1) 2 > (e 4 ) 2 e ln x = x and Power of of Power 3x + 1> e 4 Property of Inequality for Exponential Functions 3x> e 4 – 1Subtract 1 from each side.

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

22. Example 5 A.7.8732 B.8.0349 C.9.0997 D.11.232 A. Solve the equation 3 ln 6x = 12. Round to the nearest ten-thousandth.

23. Example 5 A.x > 274.66 B.x > 282.84 C.x > 286.91 D.x < 294.85 B. Solve the inequality in (4x – 2) > 7. Round to the nearest ten-thousandth.

24. Concept

25. Example 6 Solve Base e Inequalities A. SAVINGS Suppose you deposit $700 into an account paying 3% annual interest, compounded continuously. What is the balance after 8 years? A = Pe rt Continuously Compounded Interest formula = 700e (0.03)(8) Replace P with 700, r with 0.03 and t with 8. = 700e 0.24 Simplify. ≈ 889.87Use a calculator. Answer: The balance after 8 years will be $889.87.

26. Example 6 Solve Base e Inequalities B. SAVINGS Suppose you deposit $700 into an account paying 3% annual interest, compounded continuously. How long will it take for the balance in your account to reach at least $1200? The balance is at least $1200. A≥1200Write an inequality. Replace A with 700e (0.03)t. Divide each side by 700.

27. Example 6 Solve Base e Inequalities Answer: It will take about 18 years for the balance to reach at least $1200. Inverse Property of Exponents and Logarithms Divide each side by 0.03. t≥ 17.97Use a calculator.

28. Example 6 Solve Base e Inequalities C. SAVINGS Suppose you deposit $700 into an account paying 3% annual interest, compounded continuously. How much would have to be deposited in order to reach a balance of $1500 after 12 years? A = Pe rt Continuously Compounded Interest formula 1500= P ● e 0.03 ● 12 A = 1500, r = 0.003, and t = 12 Divide each side by e 0.36.

29. Example 6 Solve Base e Inequalities 1046.51≈ PUse a calculator. Answer: You need to deposit $1046.51.

30. Homework

31. Example 6 A.$46,058.59 B.$46,680.43 C.$1065.37 D.$365.37 A. SAVINGS Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously. What is the balance after 7 years?

32. Example 6 A.at least 1.27 years B.at least 7.50 years C.at least 21.22 years D.at least 124.93 years B. SAVINGS Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously. How long will it take for the balance in your account to reach at least $2500?

33. Example 6 A.$1299.43 B.$1332.75 C.$1365.87 D.$1444.60 C. SAVINGS Suppose you deposit money into an account paying 3% annual interest, compounded continuously. How much would have to be deposited in order to reach a balance of $1950 after 10 years?

34. End of the Lesson