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1. Day 26
1. Opener.
Solve for x:
x
1. 10 = 5.71
3x
2. 7e = 312

2. 2. Exponential and Logarithmic Equations.
Exponential equations: An exponential equation is an equation
containing a variable in an exponent.
Logarithmic equations: Logarithmic equations contain
logarithmic expressions and constants.
Property of Logarithms, part 2:
≠ 1, then
If x, y and a are positive numbers, a
If x = y, then log a x = log a y

3. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3

4. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1
ln 2 = ln 3

5. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3

6. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3

7. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithms

8. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithms
x ln 2 + 2 ln 2 = 2x ln 3 + ln 3

9. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithms
x ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property

10. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithms
x ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property
x ln 2 − 2x ln 3 = ln 3 − 2 ln 2

11. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithms
x ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property
Isolate terms (variable on
x ln 2 − 2x ln 3 = ln 3 − 2 ln 2 one side of the equation).

12. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithms
x ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property
Isolate terms (variable on
x ln 2 − 2x ln 3 = ln 3 − 2 ln 2 one side of the equation).
x ( ln 2 − 2 ln 3) = ln 3 − 2 ln 2

13. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithms
x ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property
Isolate terms (variable on
x ln 2 − 2x ln 3 = ln 3 − 2 ln 2 one side of the equation).
x ( ln 2 − 2 ln 3) = ln 3 − 2 ln 2 Common factor, x.

14. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithms
x ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property
Isolate terms (variable on
x ln 2 − 2x ln 3 = ln 3 − 2 ln 2 one side of the equation).
x ( ln 2 − 2 ln 3) = ln 3 − 2 ln 2 Common factor, x.
ln 3 − 2 ln 2
x=
ln 2 − 2 ln 3

15. 2. Exponential and Logarithmic Equations.
x+2 2 x+1
Example: Solve 2 =3
x+2 2 x+1 Take the log of both sides
ln 2 = ln 3
( x + 2 ) ln 2 = ( 2x + 1) ln 3 Property of Logarithms
x ln 2 + 2 ln 2 = 2x ln 3 + ln 3 Distributive property
Isolate terms (variable on
x ln 2 − 2x ln 3 = ln 3 − 2 ln 2 one side of the equation).
x ( ln 2 − 2 ln 3) = ln 3 − 2 ln 2 Common factor, x.
ln 3 − 2 ln 2
x= Divide both sides by ln2 - 2ln3
ln 2 − 2 ln 3

16. 2. Exponential and Logarithmic Equations.
Example: Solve log 4 ( x + 3) = 2

17. 2. Exponential and Logarithmic Equations.
Example: Solve log 4 ( x + 3) = 2
2
4 = x+3

18. 2. Exponential and Logarithmic Equations.
Example: Solve log 4 ( x + 3) = 2
2 Definition of Logarithm
4 = x+3

19. 2. Exponential and Logarithmic Equations.
Example: Solve log 4 ( x + 3) = 2
2 Definition of Logarithm
4 = x+3
16 = x + 3

20. 2. Exponential and Logarithmic Equations.
Example: Solve log 4 ( x + 3) = 2
2 Definition of Logarithm
4 = x+3
16 = x + 3 Simplify

21. 2. Exponential and Logarithmic Equations.
Example: Solve log 4 ( x + 3) = 2
2 Definition of Logarithm
4 = x+3
16 = x + 3 Simplify
13 = x

22. 2. Exponential and Logarithmic Equations.
Example: Solve log 4 ( x + 3) = 2
2 Definition of Logarithm
4 = x+3
16 = x + 3 Simplify
13 = x Solve for x.

23. 2. Exponential and Logarithmic Equations.
Example: Solve log 4 ( x + 3) = 2
2 Definition of Logarithm
4 = x+3
16 = x + 3 Simplify
13 = x Solve for x.
All solutions of Logarithmic equations must be checked,
because negative numbers do not have logarithms.

24. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3

25. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3

26. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms

27. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7)

28. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7) Definition of Logarithm

29. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7) Definition of Logarithm
2
8 = x − 7x

30. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7) Definition of Logarithm
2
8 = x − 7x Simplify

31. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7) Definition of Logarithm
2
8 = x − 7x Simplify
2
0 = x − 7x − 8

32. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7) Definition of Logarithm
2
8 = x − 7x Simplify
2 Write cuadratic equation in
0 = x − 7x − 8 standard form

33. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7) Definition of Logarithm
2
8 = x − 7x Simplify
2 Write cuadratic equation in
0 = x − 7x − 8 standard form
0 = ( x − 8 ) ( x + 1)

34. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7) Definition of Logarithm
2
8 = x − 7x Simplify
2 Write cuadratic equation in
0 = x − 7x − 8 standard form
0 = ( x − 8 ) ( x + 1) Solve by factoring

35. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7) Definition of Logarithm
2
8 = x − 7x Simplify
2 Write cuadratic equation in
0 = x − 7x − 8 standard form
0 = ( x − 8 ) ( x + 1) Solve by factoring
x = 8 or x = -1

36. 2. Exponential and Logarithmic Equations.
Example: Solve log 2 x + log 2 ( x − 7 ) = 3
log 2 x ( x − 7 ) = 3 Property of Logarithms
3
2 = x ( x − 7) Definition of Logarithm
2
8 = x − 7x Simplify
2 Write cuadratic equation in
0 = x − 7x − 8 standard form
0 = ( x − 8 ) ( x + 1) Solve by factoring
x = 8 or x = -1 Check!

37. 2. Exponential and Logarithmic Equations.
Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x

38. 2. Exponential and Logarithmic Equations.
Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x
4x − 3
log ( 2x − 1) = log
x

39. 2. Exponential and Logarithmic Equations.
Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x
4x − 3
log ( 2x − 1) = log Property of Logarithms
x

40. 2. Exponential and Logarithmic Equations.
Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x
4x − 3
log ( 2x − 1) = log Property of Logarithms
x
4x − 3
2x − 1 =
x

41. 2. Exponential and Logarithmic Equations.
Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x
4x − 3
log ( 2x − 1) = log Property of Logarithms
x
4x − 3
2x − 1 = Property of Logarithms
x

42. 2. Exponential and Logarithmic Equations.
Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x
4x − 3
log ( 2x − 1) = log Property of Logarithms
x
4x − 3
2x − 1 = Property of Logarithms
x
Multiply both sides by x

43. 2. Exponential and Logarithmic Equations.
Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x
4x − 3
log ( 2x − 1) = log Property of Logarithms
x
4x − 3
2x − 1 = Property of Logarithms
x
x(2x - 1) = 4x - 3 Multiply both sides by x

44. 2. Exponential and Logarithmic Equations.
Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x
4x − 3
log ( 2x − 1) = log Property of Logarithms
x
4x − 3
2x − 1 = Property of Logarithms
x
x(2x - 1) = 4x - 3 Multiply both sides by x
Distributive property

45. 2. Exponential and Logarithmic Equations.
Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x
4x − 3
log ( 2x − 1) = log Property of Logarithms
x
4x − 3
2x − 1 = Property of Logarithms
x
x(2x - 1) = 4x - 3 Multiply both sides by x
2x2 - x = 4x - 3 Distributive property

46. 2. Exponential and Logarithmic Equations.
Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x
4x − 3
log ( 2x − 1) = log Property of Logarithms
x
4x − 3
2x − 1 = Property of Logarithms
x
x(2x - 1) = 4x - 3 Multiply both sides by x
2x2 - x = 4x - 3 Distributive property
2x2 - 5x + 3 = 0

47. 2. Exponential and Logarithmic Equations.
Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x
4x − 3
log ( 2x − 1) = log Property of Logarithms
x
4x − 3
2x − 1 = Property of Logarithms
x
x(2x - 1) = 4x - 3 Multiply both sides by x
2x2 - x = 4x - 3 Distributive property
Write quadratic equation in
2x2 - 5x + 3 = 0 standard form

48. 2. Exponential and Logarithmic Equations.
Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x
4x − 3
log ( 2x − 1) = log Property of Logarithms
x
4x − 3
2x − 1 = Property of Logarithms
x
x(2x - 1) = 4x - 3 Multiply both sides by x
2x2 - x = 4x - 3 Distributive property
Write quadratic equation in
2x2 - 5x + 3 = 0 standard form
(2x - 3)(x - 1) = 0

49. 2. Exponential and Logarithmic Equations.
Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x
4x − 3
log ( 2x − 1) = log Property of Logarithms
x
4x − 3
2x − 1 = Property of Logarithms
x
x(2x - 1) = 4x - 3 Multiply both sides by x
2x2 - x = 4x - 3 Distributive property
Write quadratic equation in
2x2 - 5x + 3 = 0 standard form
(2x - 3)(x - 1) = 0 Solve by factoring

50. 2. Exponential and Logarithmic Equations.
Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x
4x − 3
log ( 2x − 1) = log Property of Logarithms
x
4x − 3
2x − 1 = Property of Logarithms
x
x(2x - 1) = 4x - 3 Multiply both sides by x
2x2 - x = 4x - 3 Distributive property
Write quadratic equation in
2x2 - 5x + 3 = 0 standard form
(2x - 3)(x - 1) = 0 Solve by factoring
x = 3/2 and x = 1

51. 2. Exponential and Logarithmic Equations.
Example: Solve log ( 2x − 1) = log ( 4x − 3) − log x
4x − 3
log ( 2x − 1) = log Property of Logarithms
x
4x − 3
2x − 1 = Property of Logarithms
x
x(2x - 1) = 4x - 3 Multiply both sides by x
2x2 - x = 4x - 3 Distributive property
Write quadratic equation in
2x2 - 5x + 3 = 0 standard form
(2x - 3)(x - 1) = 0 Solve by factoring
x = 3/2 and x = 1 Check!

52. Day 27
1. Exercises.

53. Day 27
1. Opener.

54. Day 30
1. Quiz 4.
1. “Quiz 4”.
2. Name.
3. Student Number.
4. Date.

55. 2. Quiz 4.
1. How long does it take to double an investment of $ 20,000.00 in a
bank paying an interest rate of 4% per year compounded monthly?
Find the value of x in the following equations. Check your answers.
2. 3x+4 = e5 x
3. log12 ( x − 7 ) = 1− log12 ( x − 3)
4. Character that maintained a robust dispute with Newton over the
priority of invention of calculus.
5. How did Evariste Galois die, two days after leaving prison, at age
21?
6. Why isn’t there a Nobel Prize in mathematics?