1. 6/3/2016 7:27 AM17.5 - Exp and Log Equations and Inequalities

2. Review - Exponents  Product:  Quotient:  Power: n + m n – m n* m 6/3/2016 7:27 AM 2 7.5 - Exp and Log Equations and Inequalities

3. Review - Logs  Product:  Quotient:  Power: 6/3/2016 7:27 AM 3 7.5 - Exp and Log Equations and Inequalities

4. Solving Exponential Equations If each equation on both sides are exponents: 1. Rewrite both sides by “log”-ing it 2. Use exponent and/or logarithmic rules 3. Solve algebraically, Round to 3 decimal places 6/3/2016 7:27 AM 4 7.5 - Exp and Log Equations and Inequalities

5. Example 1 Solve and Check: 9 8 – x = 27 x – 3 log 9 8 – x = log 27 x – 3 Rewrite both sides by “log”-ing it (8 – x) log 9 = (x – 3) log 27 Follow the log rules; Power Rules 8 – x = (1.5)(x – 3) Distribute 1.5 to x - 3 x = 5 Answer. 6/3/2016 7:27 AM 5 7.5 - Exp and Log Equations and Inequalities Use Algebra to solve 8 – x = 1.5x – 4.5 Solve for x. 8 – x = (x – 3) (1.5)

6. Example 1 (another way) Solve and Check: 9 8 – x = 27 x – 3 9 8 – x = 27 x – 3 Since 27 is a base of 3, apply it to both sides Solve for x x = 5 Answer. 6/3/2016 7:27 AM 6 7.5 - Exp and Log Equations and Inequalities Use Algebra to solve 16 – 2x = 3x –9

7. Example 2 Solve and Check: 8 x = 2 x + 6 x = 3 6/3/2016 7:27 AM 7 7.5 - Exp and Log Equations and Inequalities

8. Your Turn Solve and Check: 3 2x = 27 x = 3/2 6/3/2016 7:27 AM 8 7.5 - Exp and Log Equations and Inequalities

9. Example 3 Solve and Check: 4 x – 1 = 5 log 4 x – 1 = log 5 Rewrite both sides by “log”-ing it (x – 1) log 4 = log 5 Follow the log rules; Power Rules x – 1 ≈ 1.1610 Solve for x; Round to four decimal places x ≈ 2.1610 Answer. 6/3/2016 7:27 AM 9 7.5 - Exp and Log Equations and Inequalities Use Algebra to solve

10. Example 4 Solve and Check: 5 x – 2 = 200 x = 5.2920 6/3/2016 7:27 AM 10 7.5 - Exp and Log Equations and Inequalities

11. Your Turn Solve and Check: 3 2x – 1 = 20 x ≈ 1.8634 6/3/2016 7:27 AM 11 7.5 - Exp and Log Equations and Inequalities

12. Solving Logarithmic Equations If each equation on one side shows a log.: 1a. Rewrite the equation in exponential form 1b. Use exponent and/or logarithmic rules (including Change of Base) 2. Solve algebraically, Round to 3 decimal places 6/3/2016 7:27 AM 12 7.5 - Exp and Log Equations and Inequalities

13. Example 5 Solve : log 7 (5x + 3) = 3 7 3 = 5x + 3 Rewriting the equation in exponential form 343 = 5x + 3 Use Algebra to solve for x x = 68 Answer. 6/3/2016 7:27 AM 13 7.5 - Exp and Log Equations and Inequalities 5x = 340 Solve for x.

14. Example 6 Solve : log 3 (x – 5) = 2 x = 14 6/3/2016 7:27 AM 14 7.5 - Exp and Log Equations and Inequalities

15. Your Turn Solve : log 6 (7x – 1) = 1 x = 7 6/3/2016 7:27 AM 15 7.5 - Exp and Log Equations and Inequalities

16. Example 7 Solve : log 4 100 – log 4 (x + 1) = 1 Can this equation be written in Exponential Form? Rewrite equation using exponential form to solve x = 24 Answer. 6/3/2016 7:27 AM 16 7.5 - Exp and Log Equations and Inequalities Solve for x.; cross multiply NO Write problem using Log properties

17. Example 8 Solve : log x + log 5 = 2 x = 20 6/3/2016 7:27 AM 17 7.5 - Exp and Log Equations and Inequalities

18. Your Turn Solve : log (x + 4) - log 6 = 1 x = 56 6/3/2016 7:27 AM 18 7.5 - Exp and Log Equations and Inequalities

19. Example 9

20. Example 10

21. Example 11 -50 is undefined

22. Example 12

23. Example 13

24. Solving All Log Equations There are 3 different types of log equations. 1. At least 1 non-log term---get one log term by itself and rewrite it as an exponential equation. 2. No logs at all---take the log of both sides to bring the variable out of the exponent. 3. Every term is a log ---get one log term equal to one log term and then use the property to make M = N