Exponential and Logarithmic Equations

1. Exponential and Logarithmic Equations 7.5 - Exp and Log Equations and Inequalities

2. Review - Exponents • Product: • Quotient: • Power: n + m n – m n* m 7.5 - Exp and Log Equations and Inequalities

3. Review - Logs • Product: • Quotient: • Power: 7.5 - Exp and Log Equations and Inequalities

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

5. Example 1 Solve and Check: 98 – x=27x – 3 log 9 8 – x=log 27 x –3 Rewrite both sides by “log”-ing it Follow the log rules; Power Rules (8 – x) log 9 =(x – 3) log 27 Use Algebra to solve 8 – x = (x – 3) (1.5) Distribute 1.5 to x -3 8 – x = (1.5)x – 3 Solve for x. 8 – x = 1.5x – 4.5 x = 5 Answer. 7.5 - Exp and Log Equations and Inequalities

6. Example 1 (another way) Solve and Check: 98 – x=27x – 3 9 8 – x=27 x –3 Since 27 is a base of 3, apply it to both sides Use Algebra to solve 16 – 2x = 3x –9 Solve for x x = 5 Answer. 7.5 - Exp and Log Equations and Inequalities

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

8. Your Turn Solve and Check: 43x–1 = 8x+1 x = 5/3 7.5 - Exp and Log Equations and Inequalities

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

10. Your Turn Solve and Check: 32x–1 = 20 x ≈ 1.8634 7.5 - Exp and Log Equations and Inequalities

11. 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 4 decimal places 3. Check 7.5 - Exp and Log Equations and Inequalities

12. Example 6 Solve : log7(5x + 3) = 3 Rewriting the equation in exponential form 73 =5x + 3 343 = 5x + 3 Use Algebra to solve for x 5x = 340 Solve for x. x = 68 Answer. 7.5 - Exp and Log Equations and Inequalities

13. Your Turn Solve : log6(2x – 1) = –1 x = 7/12 7.5 - Exp and Log Equations and Inequalities

14. Example 7 Solve : log4100– log4(x + 1) = 1 NO Can this equation be written in Exponential Form? Write problem using Log properties Rewrite equation using exponential form to solve Solve for x.; cross multiply x = 24 Answer. 7.5 - Exp and Log Equations and Inequalities

15. Example 8 Solve : log12x + log12(x + 1) = 1 Why can’t x = –4? ---------------------- Plug –4 into original equation. ---------------------- The answer is undefined. x = 3 7.5 - Exp and Log Equations and Inequalities

16. Your Turn Solve for x: 1. 23x = 15 2. 7–x = 21 3. log5x 4 = 8 4. 3 = log 8 + 3log x 7.5 - Exp and Log Equations and Inequalities

17. Example 9 Suppose a bacteria culture doubles in size every hour. How many hours will it take for the number of bacteria to exceed 1,000,000? At hour 0, there is one bacterium, or 20 bacteria. At hour one, there are two bacteria, or 21 bacteria, and so on. So, at hour n there will be 2n bacteria. Solve 2n > 106 Write 1,000,000 in scientific annotation. log 2n > log 106 Take the log of both sides. 7.5 - Exp and Log Equations and Inequalities

18. 6 6 log 2 0.301 n > n > Example 9 Suppose a bacteria culture doubles in size every hour. How many hours will it take for the number of bacteria to exceed 1,000,000? nlog 2 > log 106 Use the Power of Logarithms. nlog 2 > 6 log 106 is 6. Divide both sides by log 2. Evaluate by using a calculator. n > ≈ 19.94 Round up to the next whole number. It will take about 20 hours for the number of bacteria to exceed 1,000,000. 7.5 - Exp and Log Equations and Inequalities

19. Assignment Worksheet Pg 526 8, 21 – 33 all NOT 27 Non-Calc Quiz Friday 7.5 - Exp and Log Equations and Inequalities