1. Solving Exponential and Logarithmic Equations
Solving when the exponent has a variable.

2. Rules of exponents Multiply……add Divide………subtract
Power to a power….multiply

3. Radicals become Fractional exponents:

4. Practice with exponents
See textbook pg. 292

5. Radical expressions Express the when coverting a radical
Expression with a fractional exponent.
Example:
1. What is the power?
2. what is the root?

6. Fractional exponents
Express as
Now evaluate:
= 32

7. Practice: See page 296 in textbook.do 3-6 297 20, 21 46-48, 58, 68,69
, , 58, 68,69
Do these:

8. Do now:
Express with fractional exponents:
Solve:

9. Solving when the base is a variable:
Solve the previous problem:

10. Procedure:
Isolate the exponential, raise to the reciprocal power, and solve:

11. Procedure:
Isolate the exponential, raise to the reciprocal power, and solve:

12. Example:
Isolate first!

13. Example:
Isolate first!

14. Exponential Equations
One way to solve exponential equations is to use the property that if 2 powers w/ the same base are equal, then their exponents are equal.
if bx = by, then x=y
For b>0 & b≠1

15. base 2, base 3,base 5 numbers: 32 64

16. Solve by equating exponents
43x = 8x+1
But 4 = 22 and 8 = 23
(22)3x = (23)x+1 rewrite w/ same base

17. Solve by equating exponents
43x = 8x+1
(22)3x = (23)x+1 rewrite w/ same base
26x = 23x+3 power to a power: multiply
6x = 3x+3 set exponents equal
x = 1
Check → 43*1 = 81+1
64 = 64

18. Procedure: Copy these steps 1. decide like base
2. rewrite one or both numbers as a power of the base
3. simplify the exponents
4. set exponents equal and solve
5. be sure to check!

19. Your turn!
24x = 32x-1

20. 24x = 32x-1 24x = (25)x-1 4x = 5x-5 5 = x Solution
Be sure to check your answer!!!

21. Hand this one in:
Solve and check:

22. Hand this one in:
Solve and check:

23. Graphs of exponentials
Growth and decay:
growth decay

24. Interest rate formula:
A=a0(1+r)t
Exponential growth and decay formula:
y = abx
a is the initial value
b is used to find the growth or decay rate.
x is the time
If b >1 then b - 1 is the growth rate
If b < 1 then 1 – b is the decay rate
Note: the rate is always in decimal form.

25. Interest rate
Use the formula to determine the amount of money you will have after 3 years if you invest 100 at a rate of 8%.
A=a0(1+r)t

26. Interest rate
Use the formula to determine the amount of money you will have after 3 years if you invest 100 at a rate of 8%.

27. Exponential examples Y = 300(.95)x What is the initial value?
What is the rate of decay?
How much will there be in 10 years?
Y = 20(1.3)t
What is the initial value
What is the growth rate?
How much will it be in 5 years?

28. COMPOUND INTEREST FORMULA
annual interest rate (as a decimal)
Principal (amount at start)
time (in years)
amount at the end
number of times per year that interest in compounded

29. Calculate how much you will have if
You invest 100 at 8% compounded
Quarterly after 3 years.

30. Calculate how much you will have if
You invest 100 at 8% compounded
Quarterly after 3 years.

31. Compound or CONTINUOUS growth:
The rate is changed to a decimal if not already.
Ex: population grows continuously at a rate of 2% in Allentown.
If Allentown has 10,000 people today, how many will it have 5 years
From now?

32. Compound or CONTINUOUS growth:
Ex: population grows continuously at a rate of 2% in Allentown.
If Allentown has 10,000 people today, how many will it have 5 years
From now?

33. How much will a 1000 investment be
Worth in 10 years if it is compounded continuously
at 2.5%?
If a substance decays continually at a rate of an hour,
How much will be left of a 20 gram substance in 6 hours?

34. How much will a 1000 investment be
Worth in 10 years if it is compounded continuously
At 2.5%?
If a substance decays continually at a rate of an hour,
How much will be left of a 20 gram substance in 6 hours?

35. Solving exponentials w/ variable exponents-like bases
1. decide like base
2. rewrite one or both numbers as a power of the base
3. simplify the exponents
4. set exponents equal and solve
5. be sure to check!
Example 43x = 8x+1

36. Solve by equating exponents
43x = 8x+1
(22)3x = (23)x+1 rewrite w/ same base
26x = 23x+3 power to a power: multiply
6x = 3x+3 set exponents equal
x = 1
Check → 43*1 = 81+1
64 = 64

37. When you can’t rewrite using the same base: 1. isolate the base term 2
When you can’t rewrite using the same base: 1. isolate the base term 2. “log” both sides 3. Solve for x
Example: x = 7
log 2x = log 7

38. 2x = 7 log 2x = log 7 x log 2 = log 7 x = ≈ 2.807
When you can’t rewrite using the same base, you can solve by taking a log of both sides
2x = 7
log 2x = log 7
x log 2 = log 7
x = ≈ 2.807

39. 4x = 15
“log” both sides & solve

40. 4x = 15
log 4x = log 15
x log 4 = log15
x= log 15/log 4
≈ 1.95

41. 10(2x-3) +4 = 21
Isolate the base term

42. 10(2x-3) +4 = 21
102x-3 = 17
Now solve for x….

43. 10(2x-3) +4 = 21 -4 -4 102x-3 = 17 log102x-3 = log17
102x-3 = 17
log102x-3 = log17
(2x-3)(log 10) = log 17
2x – 3 = log17/log 10
2x =( )
x= ≈ 2.115

44. 5(3)x/4 = 25
isolate and solve:

45. Solution:
Ex:

46. Solution:
Ex:

47. Newton’s Law of Cooling
The temperature T of a cooling time t (in minutes) is:
T = (T0 – TR) e-rt + TR
T0= initial temperature
TR= room temperature
r = constant cooling rate of the substance

48. You’re cooking stew. When you take it off the stove the temp. is 212°F
You’re cooking stew. When you take it off the stove the temp. is 212°F. The room temp. is 70°F and the cooling rate of the stew is r = How long will it take to cool the stew to a serving temp. of 100°?

49. So solve: 100 = (212 – 70)e-.046t +70 30 = 142e-.046t (subtract 70)
T0 = 212, TR = 70, T = r = .046
So solve:
100 = (212 – 70)e-.046t +70
30 = 142e-.046t (subtract 70)
.211 ≈ e-.046t (divide by 142)
How do you get the variable out of the exponent?

50. ln .211 ≈ ln e-.046t (take the ln of both sides) ln .211 ≈ -.046t ln e
Cooling cont.
ln .211 ≈ ln e-.046t (take the ln of both sides)
ln .211 ≈ -.046t ln e
≈ -.046t (1)
33.8 ≈ t
about 34 minutes to cool!

51. Domains of log equations
Find the domain of # 19 on page 331
Domains are important because solving logarithmic equations sometimes produces extraneous roots.

52. If logbx = logby, then x = y
Solving Log Equations
To solve use the property for logs w/ the same base:
+ #’s b,x,y & b≠1
If logbx = logby, then x = y

53. Solve by decompressing
log3(5x-1) = log3(x+7)
Solve by decompressing

54. 5x – 1 = x + 7 5x = x + 8 4x = 8 x = 2 and check
log3(5x-1) = log3(x+7)
5x – 1 = x + 7
5x = x + 8
4x = 8
x = 2 and check
log3(5*2-1) = log3(2+7)
log39 = log39

55. When you can’t rewrite both sides as logs w/ the same base exponentiate each side
b>0 & b≠1
if x = y, then bx = by

56. log5(3x + 1) = 2
use base 5 on both sides

57. 3x+1 = 25 x = 8 and check log5(3x + 1) = 2 (3x+1) = 52
Because the domain of log functions doesn’t include all reals, you should check for extraneous solutions

58. log5x + log(x+1)=2
Decompress and exponentiate

59. log5x + log(x+1)=2 10log5x +5x = 102
log (5x)(x+1) = (product property)
log (5x2 + 5x) = 2
10log5x +5x = 102
5x2 + 5x = 100
x2 + x - 20 = (subtract 100 and divide by 5)
(x+5)(x-4) = x=-5, x=4
graph and you’ll see 4=x is the only solution 2

60. One More! log2x + log2(x-7) = 3
Solve and check:

61. One More! log2x + log2(x-7) = 3
log2x(x-7) = 3
log2 (x2- 7x) = 3
2log2(x -7x) = 23
x2 – 7x = 8
x2 – 7x – 8 = 0
(x-8)(x+1)=0
x=8 x= -1 2

62. Page 331 # 38
Solve and ck.
log(x-2) + log(x+5) =2log 3

63. Solution: log(x-2) + log(x+5) =2log 3 (x-2)(x+5)=9 x2 + 3x -19 = 0
(the neg. answer is extraneous)

64. Assignment
Hw pg , odds