1. Solving Exponential and Logarithmic Equations Section 3.4

2. Objectives Solve simple exponential and logarithmic equations. Solve more complicated exponential equations. Solve more complicated logarithmic equations.

3. Solving Exponential and Logarithmic Equations There are two basic strategies for solving exponential or logarithmic equations. The first is based on the One-to-One properties and the second is based on the Inverse Properties.

4. Strategies

5. Solving Exponential Equations Equations that involve terms of the form a x, a > 0, a  1 are called exponential equations. Property for Solving Exponential Equations If a u = a v, then u = v. To use this property, each side of the equality must be written with the same base.

6. Solving Exponential Equations Steps for Solving Exponential Equations of the Form a u = a v Step 1: Use the Laws of Exponents to write both sides of the equation with the same base. Step 2: Set the exponential expressions on each side of the equation equal to each other. Step 3: Solve the equation resulting from Step 2. Step 4: Check. Verify your solution(s).

7. Solving Exponential Equations with Like Bases Example: Solve: 64 = 2 6 The bases must be the same. Set the exponential expressions equal to each other. Use the Law of Exponents. Check:

8. Exponential Equations with Like Bases Example - One exponential expression. 1. Isolate the exponential expression and rewrite the constant in terms of the same base. 2. Set the exponents equal to each other (drop the bases) and solve the resulting equation.

9. Your Turn: Remember a negative exponent is simply another way of writing a fraction The bases are now the same so set the exponents equal. Exponential Equations with Like Bases

10. Solve 3 -3 = 3 2x – 3 = 2x Exponential Equations with Like Bases Your Turn:

11. Solve 9 2x+1 = 81 9 2x+1 = 9 2 2x + 1 = 2 2x = 1 x = ½ Exponential Equations with Like Bases Your Turn:

12. EQUATIONS OF THE FORM a f(x) = a g(x)

13. Exponential Equations of the Form a f(x) = a g(x) with Like Bases Example - Two exponential expressions. 1. Isolate the exponential expressions on either side of the =. We then rewrite the 2nd expression in terms of the same base as the first. 2. Set the exponents equal to each other (drop the bases) and solve the resulting equation.

14. Solve 4 3x-6 = 32 2x (2 2 ) 3x-6 = (2 5 ) 2x (2 2 ) 3x-6 = 2 10x 2 6x-12 = 2 10x 6x – 12 = 10x – 12 = 4x x = – 3 Exponential Equations of the Form a f(x) = a g(x) with Like Bases Your Turn:

15. Your Turn: : Exponential Equations of the Form a f(x) = a g(x) with Like Bases

16. Your Turn: Exponential Equations of the Form a f(x) = a g(x) with Like Bases

17. Your Turn: x = -1/2 x = 3/4

18. Natural Exponential Equations Example: –e 2x+1 = e -(x-4) 2x+1 = -(x-4) 2x+1 = -x+4 3x+1 = 4 3x=3 x=1

19. Solving Exponential Equations Since logs are the inverse of exponentials, logs can be used to solve exponential equations.

20. Exponential Equations with Different Bases The Exponential Equations below contain exponential expressions whose bases cannot be rewritten as the same rational number. The solutions are irrational numbers, we will need to use a log function to evaluate them. or

21. Exponential Equations with Different Bases Example #1 - One exponential expression. 1. Isolate the exponential expression. 3. Use the log rule that lets you rewrite the exponent as a multiplier. 2. Take the log (log or ln) of both sides of the equation.

22. Exponential Equations with Different Bases Example #1 - One exponential expression. 4. Isolate the variable.

23. Exponential Equations with Different Bases Example #2 - Two exponential expressions. 1. The exponential expressions are already isolated. 3. Use the log rule that lets you rewrite the exponent as a multiplier on each side.. 2. Take the log (log or ln) of both sides of the equation.

24. Exponential Equations with Different Bases Example #2 - Two exponential expressions. 4. To isolate the variable, we need to combine the ‘x’ terms, then factor out the ‘x’ and divide.

25. Methods of Solving Logarithmic Equations 1.Algebraic solution a)Equal base method b)Inverse operation method

26. Solving Log Equations – Equal Base Method To solve use the property for logs w/ the same base: If log b x = log b y, then x = y

27. Use property for logarithms with the same base. 5 x = x + 8 Solving a Logarithmic Equation Solve log 3 (5 x – 1) = log 3 (x + 7). 5 x – 1 = x + 7 x = 2 The solution is 2. S OLUTION log 3 (5 x – 1) = log 3 (x + 7) Write original equation. Add 1 to each side. Solve for x. C HECK Check the solution by substituting it into the original equation. log 3 (5 x – 1) = log 3 (x + 7) log 3 9 = log 3 9 Solution checks. log 3 (5 · 2 – 1) = log 3 (2 + 7) ? Write original equation. Substitute 2 for x.

28. Inverse Operation Method When you can’t rewrite the equation using the same base, you can solve by taking exponentiating both sides 1.Simplify log expressions using the properties of logs (condense to single log expressions). 2.Solve equation using inverse operations in the inverse order (exponentiating both sides to undo the logs). 3.Check solution.

29. log 5 (3x + 1) = 2 Solving a Logarithmic Equation Solve log 5 (3x + 1) = 2. 5 = 5 2 log 5 (3x + 1) 3x + 1 = 25 x = 8 The solution is 8. S OLUTION Write original equation. Exponentiate each side using base 5. b = x log b x Solve for x. log 5 (3x + 1) = 2 log 5 (3 · 8 + 1) = 2 ? log 5 25 = 2 ? 2 = 2 Solution checks. C HECK Check the solution by substituting it into the original equation. Simplify. Substitute 8 for x. Write original equation.

30. Because the domain of a logarithmic function generally does not include all real numbers, you should be sure to check for extraneous solutions of logarithmic equations. You can do this algebraically or graphically. Checking for Extraneous Solutions

31. Solve log 5 x + log (x – 1) = 2. Check for extraneous solutions. log [ 5 x (x – 1)] = 2 10 = 10 2 log (5 x 2 – 5 x) 5 x 2 – 5 x = 100 x 2 – x – 20 = 0 (x – 5 )(x + 4) = 0 x = 5 or x = – 4 S OLUTION log 5 x + log (x – 1) = 2 Write original equation. Product property of logarithms. Exponentiate each side using base 10. 10 log x = x Write in standard form. Factor. Zero product property Checking for Extraneous Solutions

32. y x The solutions appear to be 5 and – 4. However, when you check these in the original equation or use a graphic check as shown below, you can see that x = 5 is the only solution. S OLUTION log 5 x + log (x – 1) = 2 Check for extraneous solutions. x = 5 or x = – 4 Zero product property Checking for Extraneous Solutions y = 2 y = log 5 x + log (x – 1)

33. More Logarithmic Equations In a Logarithmic Equation, the variable can be inside the log function or inside the base of the log. There may be one log term or more than one. For example …

34. Logarithmic Equations Example 1 - Variable inside the log function. 1. Isolate the log expression. 2. Rewrite the log equation as an exponential equation and solve for ‘x’.

35. Logarithmic Equations Example 2 - Variable inside the log function, two log expressions. 1. To isolate the log expression, we 1st must use the log property to combine a difference of logs. 2. Rewrite the log equation as an exponential equation (here, the base is ‘e’). 3. To solve for ‘x’ we must distribute the ‘e’ and then collect the ‘x’ terms together and factor out the ‘x’ and divide.

36. Logarithmic Equations Example 3 - Variable inside the base of the log. 1. Rewrite the log equation as an exponential equation. 2. Solve the exponential equation.

37. One More! log 2 x + log 2 (x-7) = 3 log 2 x(x-7) = 3 log 2 (x 2 - 7x) = 3 2 log 2 (x 2 - 7x) = 2 3 x 2 – 7x = 8 x 2 – 7x – 8 = 0 (x-8)(x+1)=0 x=8 x= -1

38. YOUR TURN: Additional Logarithmic and Exponential Equation Problems

42. Both terms are undefined. Check x = 3. Solution set {x | x = 3}.

44. Equation of Quadratic Type

45. No solution. Solution x = 0. Solution set {x | x =0}.

47. x

48. Assignment Pg. 221 – 224; #1 – 7 odd, 17 -49 odd, 61 – 67 odd, 85 – 105 odd