This document contains examples and explanations of exponential equations and logarithmic equations. It shows how exponential equations can be rewritten as logarithmic equations by taking the logarithm of both sides. It also explains how logarithmic equations with different bases can be converted using the change of base formula. Examples are provided to demonstrate solving exponential and logarithmic equations. The document also contains an example using an exponential growth model to project future population growth in the US.

1. Section 6-6
Solving Exponential Equations

2. Warm-up
GIVE FIVE PAIRS (X, Y) OF NUMBERS FOR WHICH YOU CAN DETERMINE EXACT SOLUTIONS TO
THE EQUATION:
y
15 = x

3. Warm-up
GIVE FIVE PAIRS (X, Y) OF NUMBERS FOR WHICH YOU CAN DETERMINE EXACT SOLUTIONS TO
THE EQUATION:
y
15 = x
HOW ABOUT (1, 0), (15, 1), (225, 2), (3375, 3), ETC.

4. Warm-up
GIVE FIVE PAIRS (X, Y) OF NUMBERS FOR WHICH YOU CAN DETERMINE EXACT SOLUTIONS TO
THE EQUATION:
y
15 = x
HOW ABOUT (1, 0), (15, 1), (225, 2), (3375, 3), ETC.
ALL OF THESE ANSWERS ALSO FIT THE EQUATION:
y = log15 x

5. Exponential Equation

6. Exponential Equation
AN EQUATION THAT HAS A VARIABLE IN THE EXPONENT

7. Blog Question

8. Blog Question
HOW ARE EXPONENTIAL EQUATIONS AND LOGARITHMIC EQUATIONS
RELATED?

9. Example 1
IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.

10. Example 1
IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.
REWRITING AS A LOGARITHM DOESN’T SEEM TO DO ANYTHING FOR US.

11. Example 1
IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.
REWRITING AS A LOGARITHM DOESN’T SEEM TO DO ANYTHING FOR US.
log2 3 = t

12. Example 1
IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.
REWRITING AS A LOGARITHM DOESN’T SEEM TO DO ANYTHING FOR US.
log2 3 = t
WHAT ABOUT TAKING THE LOGARITHM OF EACH SIDE OF THE EXPONENTIAL?

13. Example 1
IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.
REWRITING AS A LOGARITHM DOESN’T SEEM TO DO ANYTHING FOR US.
log2 3 = t
WHAT ABOUT TAKING THE LOGARITHM OF EACH SIDE OF THE EXPONENTIAL?
t
2 =3

14. Example 1
IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.
REWRITING AS A LOGARITHM DOESN’T SEEM TO DO ANYTHING FOR US.
log2 3 = t
WHAT ABOUT TAKING THE LOGARITHM OF EACH SIDE OF THE EXPONENTIAL?
t
2 =3
t
log 2 = log 3

15. Example 1
IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.
REWRITING AS A LOGARITHM DOESN’T SEEM TO DO ANYTHING FOR US.
log2 3 = t
WHAT ABOUT TAKING THE LOGARITHM OF EACH SIDE OF THE EXPONENTIAL?
t
2 =3
t
log 2 = log 3
t log 2 = log 3

16. Example 1
IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.
REWRITING AS A LOGARITHM DOESN’T SEEM TO DO ANYTHING FOR US.
log2 3 = t
WHAT ABOUT TAKING THE LOGARITHM OF EACH SIDE OF THE EXPONENTIAL?
t
2 =3
t
log 2 = log 3
t log 2 = log 3
log 2 log 2

17. Example 1
IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.
REWRITING AS A LOGARITHM DOESN’T SEEM TO DO ANYTHING FOR US.
log2 3 = t
WHAT ABOUT TAKING THE LOGARITHM OF EACH SIDE OF THE EXPONENTIAL?
t
2 =3
t
log 2 = log 3
t log 2 = log 3
log 2 log 2
log3
t = log2

18. Example 1
IF 2t = 3, FIND t TO THE NEAREST HUNDREDTH.
REWRITING AS A LOGARITHM DOESN’T SEEM TO DO ANYTHING FOR US.
log2 3 = t
WHAT ABOUT TAKING THE LOGARITHM OF EACH SIDE OF THE EXPONENTIAL?
t
2 =3
t
log 2 = log 3
t log 2 = log 3
log 2 log 2
t = log2 ≈ 1.58
log3

19. Change of Base
Theorem

20. Change of Base
Theorem
FOR ALL VALUES OF a, b, AND c WHERE THE LOGS EXIST:
logc a ln a
logb a = =
logc b ln b

21. Change of Base
Theorem
FOR ALL VALUES OF a, b, AND c WHERE THE LOGS EXIST:
logc a ln a
logb a = =
logc b ln b
THIS MEANS WE CAN SOLVE ANY LOGARITHM USING COMMON LOGS OR
NATURAL LOGS

22. Example 2
EVALUATE:

23. Example 2
EVALUATE:
log17 34

24. Example 2
EVALUATE:
log17 34
log 34
=
log17

25. Example 2
EVALUATE:
log17 34
log 34
=
log17
≈ 1.24

26. Example 2
EVALUATE:
log17 34 log17 34
log 34
=
log17
≈ 1.24

27. Example 2
EVALUATE:
log17 34 log17 34
log 34 ln 34
= =
log17 ln17
≈ 1.24

28. Example 2
EVALUATE:
log17 34 log17 34
log 34 ln 34
= =
log17 ln17
≈ 1.24 ≈ 1.24

29. Example 3
THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND
WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?

30. Example 3
THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND
WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?
rt
A (t ) = Pe

31. Example 3
THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND
WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?
rt
A (t ) = Pe
.01t
300,000,000 = 250,000,000e

32. Example 3
THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND
WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?
rt
A (t ) = Pe
.01t
300,000,000 = 250,000,000e
250,000,000 250,000,000

33. Example 3
THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND
WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?
rt
A (t ) = Pe
.01t
300,000,000 = 250,000,000e
250,000,000 250,000,000
.01t
1.2 = e

34. Example 3
THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND
WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?
rt
A (t ) = Pe
.01t
300,000,000 = 250,000,000e
250,000,000 250,000,000
.01t
1.2 = e
.01t
ln1.2 = ln e

35. Example 3
THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND
WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?
rt
A (t ) = Pe
.01t
300,000,000 = 250,000,000e
250,000,000 250,000,000
.01t
1.2 = e
.01t
ln1.2 = ln e
ln1.2 = .01t

36. Example 3
THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND
WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?
rt
A (t ) = Pe
.01t
300,000,000 = 250,000,000e
250,000,000 250,000,000
.01t
1.2 = e
.01t
ln1.2 = ln e
ln1.2 = .01t t= ln1.2
.01

37. Example 3
THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND
WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?
rt
A (t ) = Pe
.01t
300,000,000 = 250,000,000e
250,000,000 250,000,000
.01t
1.2 = e
.01t
ln1.2 = ln e
ln1.2 = .01t t= ln1.2
.01
t ≈ 18

38. Example 3
THE POPULATION OF THE US REACHED 250 MILLION IN 1990 AND WAS
GROWING AT ABOUT 1% PER YEAR. USE THE FORMULA A(t) = Pert TO FIND
WHEN, AT THIS GROWTH RATE, THE POPULATION WOULD REACH 300
MILLION. HOW DOES THIS PROJECTION HOLD UP?
rt
A (t ) = Pe
.01t
300,000,000 = 250,000,000e
250,000,000 250,000,000
.01t
1.2 = e
.01t
ln1.2 = ln e
ln1.2 = .01t t= ln1.2
.01
t ≈ 18 YEARS

39. Example 4
IN SECTION 6-2, THE FORMULA T(p) = 10p0.585 IS GIVEN FOR THE
APPROXIMATE TIME T(p) IT TAKES TO MICROWAVE p PORTIONS OF FOOD,
ASSUMING THAT DOUBLING THE NUMBER OF PORTIONS REQUIRES COOKING
TIME TO BE MULTIPLIED BY 1.5. THE FORMULA WAS SAID TO HAVE BEEN
DEDUCED FROM THE FACT THAT 1 PORTION REQUIRED 10 MINUTES TO COOK
AND 2 PORTIONS REQUIRED 15 MINUTES. SHOW HOW THIS FORMULA WAS
CALCULATED FROM THE ASSUMPTION THAT THERE WAS A MODEL OF THE
FORM T(p) = 10px.
x x
T (1) = 10i1 = 10 T (2) = 10i2 = 15

40. Example 4
IN SECTION 6-2, THE FORMULA T(p) = 10p0.585 IS GIVEN FOR THE
APPROXIMATE TIME T(p) IT TAKES TO MICROWAVE p PORTIONS OF FOOD,
ASSUMING THAT DOUBLING THE NUMBER OF PORTIONS REQUIRES COOKING
TIME TO BE MULTIPLIED BY 1.5. THE FORMULA WAS SAID TO HAVE BEEN
DEDUCED FROM THE FACT THAT 1 PORTION REQUIRED 10 MINUTES TO COOK
AND 2 PORTIONS REQUIRED 15 MINUTES. SHOW HOW THIS FORMULA WAS
CALCULATED FROM THE ASSUMPTION THAT THERE WAS A MODEL OF THE
FORM T(p) = 10px.
x x
T (1) = 10i1 = 10 T (2) = 10i2 = 15
x
1 =1

41. Example 4
IN SECTION 6-2, THE FORMULA T(p) = 10p0.585 IS GIVEN FOR THE
APPROXIMATE TIME T(p) IT TAKES TO MICROWAVE p PORTIONS OF FOOD,
ASSUMING THAT DOUBLING THE NUMBER OF PORTIONS REQUIRES COOKING
TIME TO BE MULTIPLIED BY 1.5. THE FORMULA WAS SAID TO HAVE BEEN
DEDUCED FROM THE FACT THAT 1 PORTION REQUIRED 10 MINUTES TO COOK
AND 2 PORTIONS REQUIRED 15 MINUTES. SHOW HOW THIS FORMULA WAS
CALCULATED FROM THE ASSUMPTION THAT THERE WAS A MODEL OF THE
FORM T(p) = 10px.
x x
T (1) = 10i1 = 10 T (2) = 10i2 = 15
x
1 =1
log1= 0

42. Example 4
IN SECTION 6-2, THE FORMULA T(p) = 10p0.585 IS GIVEN FOR THE
APPROXIMATE TIME T(p) IT TAKES TO MICROWAVE p PORTIONS OF FOOD,
ASSUMING THAT DOUBLING THE NUMBER OF PORTIONS REQUIRES COOKING
TIME TO BE MULTIPLIED BY 1.5. THE FORMULA WAS SAID TO HAVE BEEN
DEDUCED FROM THE FACT THAT 1 PORTION REQUIRED 10 MINUTES TO COOK
AND 2 PORTIONS REQUIRED 15 MINUTES. SHOW HOW THIS FORMULA WAS
CALCULATED FROM THE ASSUMPTION THAT THERE WAS A MODEL OF THE
FORM T(p) = 10px.
x x
T (1) = 10i1 = 10 T (2) = 10i2 = 15
x
1 =1
log1= 0
THIS WON’T HELP

43. Example 4
IN SECTION 6-2, THE FORMULA T(p) = 10p0.585 IS GIVEN FOR THE
APPROXIMATE TIME T(p) IT TAKES TO MICROWAVE p PORTIONS OF FOOD,
ASSUMING THAT DOUBLING THE NUMBER OF PORTIONS REQUIRES COOKING
TIME TO BE MULTIPLIED BY 1.5. THE FORMULA WAS SAID TO HAVE BEEN
DEDUCED FROM THE FACT THAT 1 PORTION REQUIRED 10 MINUTES TO COOK
AND 2 PORTIONS REQUIRED 15 MINUTES. SHOW HOW THIS FORMULA WAS
CALCULATED FROM THE ASSUMPTION THAT THERE WAS A MODEL OF THE
FORM T(p) = 10px.
x x
T (1) = 10i1 = 10 T (2) = 10i2 = 15
x x
1 =1 2 = 1.5
log1= 0
THIS WON’T HELP

44. Example 4
IN SECTION 6-2, THE FORMULA T(p) = 10p0.585 IS GIVEN FOR THE
APPROXIMATE TIME T(p) IT TAKES TO MICROWAVE p PORTIONS OF FOOD,
ASSUMING THAT DOUBLING THE NUMBER OF PORTIONS REQUIRES COOKING
TIME TO BE MULTIPLIED BY 1.5. THE FORMULA WAS SAID TO HAVE BEEN
DEDUCED FROM THE FACT THAT 1 PORTION REQUIRED 10 MINUTES TO COOK
AND 2 PORTIONS REQUIRED 15 MINUTES. SHOW HOW THIS FORMULA WAS
CALCULATED FROM THE ASSUMPTION THAT THERE WAS A MODEL OF THE
FORM T(p) = 10px.
x x
T (1) = 10i1 = 10 T (2) = 10i2 = 15
x x
1 =1 2 = 1.5
log1= 0 log2 1.5 = x
THIS WON’T HELP

45. Example 4
IN SECTION 6-2, THE FORMULA T(p) = 10p0.585 IS GIVEN FOR THE
APPROXIMATE TIME T(p) IT TAKES TO MICROWAVE p PORTIONS OF FOOD,
ASSUMING THAT DOUBLING THE NUMBER OF PORTIONS REQUIRES COOKING
TIME TO BE MULTIPLIED BY 1.5. THE FORMULA WAS SAID TO HAVE BEEN
DEDUCED FROM THE FACT THAT 1 PORTION REQUIRED 10 MINUTES TO COOK
AND 2 PORTIONS REQUIRED 15 MINUTES. SHOW HOW THIS FORMULA WAS
CALCULATED FROM THE ASSUMPTION THAT THERE WAS A MODEL OF THE
FORM T(p) = 10px.
x x
T (1) = 10i1 = 10 T (2) = 10i2 = 15
x x
1 =1 2 = 1.5
log1= 0 log2 1.5 = x
log1.5
THIS WON’T HELP =
log 2

46. Example 4
IN SECTION 6-2, THE FORMULA T(p) = 10p0.585 IS GIVEN FOR THE
APPROXIMATE TIME T(p) IT TAKES TO MICROWAVE p PORTIONS OF FOOD,
ASSUMING THAT DOUBLING THE NUMBER OF PORTIONS REQUIRES COOKING
TIME TO BE MULTIPLIED BY 1.5. THE FORMULA WAS SAID TO HAVE BEEN
DEDUCED FROM THE FACT THAT 1 PORTION REQUIRED 10 MINUTES TO COOK
AND 2 PORTIONS REQUIRED 15 MINUTES. SHOW HOW THIS FORMULA WAS
CALCULATED FROM THE ASSUMPTION THAT THERE WAS A MODEL OF THE
FORM T(p) = 10px.
x x
T (1) = 10i1 = 10 T (2) = 10i2 = 15
x x
1 =1 2 = 1.5
log1= 0 log2 1.5 = x
log1.5
THIS WON’T HELP =
log 2
≈ .585

47. Homework

48. Homework
P. 406 #1-20