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SOLVING LOGARITHMIC EQUATIONS. STEPS: 1.Get the LOG, LN, or e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential.

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Presentation on theme: "SOLVING LOGARITHMIC EQUATIONS. STEPS: 1.Get the LOG, LN, or e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential."— Presentation transcript:

1 SOLVING LOGARITHMIC EQUATIONS

2 STEPS: 1.Get the LOG, LN, or e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

3 Solving Logarithmic Equations Example 1: Solve: log 7 7 = x STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

4 Solving Logarithmic Equations Example 1: Solve: log 7 7 = x STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

5 Solving logarithmic equations Example 2: Solve: log 2 8 = x STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

6 Solving logarithmic equations Example 2: Solve: log 2 8 = x STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

7 Solving logarithmic equations Example 3: Solve: log 1000 = x STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

8 Solving logarithmic equations Example 3: Solve: log 1000 = x STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

9 Solving logarithmic equations Example 4: Solve: ln 1 = x STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

10 Solving logarithmic equations Example 4: Solve: ln 1 = x STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

11 Solving logarithmic equations Example 5: Solve: log 3 x = 4 STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

12 Solving logarithmic equations Example 5: Solve: log 3 x = 4 STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

13 Solving logarithmic equations Example 6: Solve: log 5 x= 1 STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

14 Solving logarithmic equations Example 6: Solve: log 5 x= 1 STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

15 Solving logarithmic equations Example 7: Solve: log x= 4 STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

16 Solving logarithmic equations Example 7: Solve: log x= 4 STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

17 Solving logarithmic equations Example 8: Solve: log 2 (x – 1)= 3 STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

18 Solving logarithmic equations Example 8: Solve: log 2 (x – 1)= 3 STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

19 Solving logarithmic equations Example 9: Solve: ln (x + 1)= 4 STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

20 Solving logarithmic equations Example 9: Solve: ln (x + 1)= 4 STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

21 Solving logarithmic equations Example 10: Solve: log 3 x + 4 = 6 STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

22 Solving logarithmic equations Example 10: Solve: log 3 x + 4 = 6 STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

23 Solving logarithmic equations Example 11: Solve: 5e x = 30 STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

24 Solving logarithmic equations Example 11: Solve: 5e x = 30 STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

25 Solving logarithmic equations Example 12: Solve: e x − 1 = 10 STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

26 Solving logarithmic equations Example 12: Solve: e x − 1 = 10 STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

27 Solving logarithmic equations Example 13: Solve: 3 x + 1 = 5 STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

28 Solving logarithmic equations Example 13: Solve: 3 x + 1 = 5 STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x.

29 ON YOUR OWN 1. log 8 64 = x 2. log 5 x= 1 STEPS: 1. Get the LOG, LN, OR e expression alone. 2. Convert to the opposite form. Logarithmic ---> Exponential Exponential ---> Logarithmic 3. Simplify. 4. Solve for x. Solve: 3. ln x = 2 4. log 4 (x – 5)= 2 5. ln (x + 2) = 1 6. e x − 1 = 10 7. 3 x + 1 = 5 8. 5 x - 3 = 125

30 Evaluating Logarithms STEPS: 1. Set = x. 2. Use B.O.B. to rewrite as an exponential expression. 3. Solve for x (use mental math). 1. log 4 162. log 8 1 3. log 1 4. log 2 8

31 USING THE GRAPHING CALCULATOR TO EVALUATE LOGS

32 STEPS: 1. Press the MATH button. 2. Click on 9: logBASE(. 3. Type in the base and answer. 4. Press the ENTER button. USING THE GRAPHING CALCULATOR TO SIMPLIFY LOGS

33 STEPS: 1. Press the MATH button. 2. Click on 9: logBASE(. 3. Type in the base and answer. 4. Press the ENTER button. 1. log 4 162. log 8 1 3. log 1 4. log 2 8 Evaluating Logarithms

34 STEPS: 1. Press the MATH button. 2. Click on 9: logBASE(. 3. Type in the base and answer. 4. Press the ENTER button. 1. log 4 162. log 8 1 3. log 1 4. log 2 8 Evaluating Logarithms

35 USING LOGS TO SOLVE EXPONENTIAL WORD PROBLEMS

36 REMEMBER … A car costs $35,000 new but depreciates 12% per year each succeeding year. What will the car cost in 5 years? A = P(1 + r) t A = P(1 − r) t

37 REMEMBER … A car costs $35,000 new but depreciates 12% per year each succeeding year. What will the car cost in 5 years? A = P(1 + r) t A = P(1 − r) t

38 Solving for t. A = P(1 + r) t A = P(1 − r) t A city’s population is 65,000. It is predicted that the population will increase by 3% every year. How long will it take for the population to reach 80,000?

39 Solving for t. A = P(1 + r) t A = P(1 − r) t A city’s population is 65,000. It is predicted that the population will increase by 3% every year. How long will it take for the population to reach 80,000?

40 Solving for t. A = P(1 + r) t A = P(1 − r) t You invest $1,250 in an account that pays 5% interest annually. In how many years will your investment grow to be $2,000?

41 Solving for t. A = P(1 + r) t A = P(1 − r) t You invest $1,250 in an account that pays 5% interest annually. In how many years will your investment grow to be $2,000?

42 IN-CLASS ASSIGNMENT  p. 273 (#s 14-16, 19-20,)  P. 274 (#s 28-33) COMPLETE EXPLORATIONS WORKBOOK

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