dgriffinresources.netdgriffinresources.net/math102ch7book.pdf · 7% 1 16.- $2100 6.5% 2 17.- $1300...

Exponential and Logarithmic expressions

E - 1

7.1 Inverse Functions

Determine whether the function described below is one-to-one. If it is, write its inverse. If it is not, indicate the reason.

1.- ( ) ( ) ( ) ( ){ }3,1 , 4,2 , 2,0 , 5,3−

2.- ( ) ( ) ( ) ( ){ }2,3 , 6,5 , 9,2 , 1,3−

3.- ( ) ( ) ( ) ( ){ }4,1 , 1,4 , 3,2 , 2,3

4.- ( ) ( ) ( ) ( ){ }5,5 , 4,6 , 3,0 , 2, 4− − −

Find the inverse of the following one-to-one functions:

5.- ( ) 2f x x= +

6.- ( ) 6f x x= −

7.- ( ) 2 1 f x x= +

8.- ( ) 4 7 f x x= −

9.- ( ) 32xf x = +

10.- ( ) 13 6xf x = −

11.- ( ) 12xf x −=

12.- ( ) 65xf x +=

Alternative phrasing: Solve for x in the following equalities: 5.- 2y x= +

6.- 6y x= −

7.- 2 1 y x= +

8.- 4 7 y x= −

9.- 3

2xy = +


E - 2

10.- 13 6xy = −

11.- 12xy −=

12.- 65xy +=


E - 3

7.2 Exponential espressions and equations Evaluate the following exponential expressions for the requested value.

1.- 3x for 4x =

2.- 2 5x + for 3x =

3.- 4 17x − for 2x =

4.- 1 12

x⎛ ⎞ +⎜ ⎟⎝ ⎠

for 2x =

5.- 19

x⎛ ⎞⎜ ⎟⎝ ⎠

for 12

x =

6.- 8x for 13

x = −

7.- 1 24

x⎛ ⎞ −⎜ ⎟⎝ ⎠

for 1x = −

8.- 2 13 5

x⎛ ⎞ −⎜ ⎟⎝ ⎠

for 0x =

9.- 3 12 3

x⎛ ⎞ +⎜ ⎟⎝ ⎠

for 1x = −

Write the given expression with the requested base:

10.-

25 with base 5

11.-

16 with base 2

12.-

81 with base 3

13.-

64 with base 2

14.-

64 with base 4

15.-

49 with base 3 16.-

27y with base 3 17.-

8x with base 2

18.-

116x− with base 4 19.- 216x+ with base 2


E - 4

Using the Compound Interest formula 1 n trA P

n

⋅⎛ ⎞= +⎜ ⎟⎝ ⎠ find the accumulated

amount for a deposit of:

Principal at annual rate at the end of

compounded Answer

20.- $2000 4.6 % 3 years yearly $2288.89 21.- $3500 2.7 % 4 years monthly $3898.69 22.- $4600 3.8 % 1 year quarterly $4777.31 23.- $2500 5.3 % 2 years semi-annually $2775.72 24.- $3100 2.1 % 3 years daily $3301.58

Exponentials and Logs – Exponential Equations

Solve for x in the following exponential equations: (Hint: Write the expressions on both sides with the same base, if possible, and set the exponents equal. Then solve for the unknown.)

25.- 5 125x =

26.- 9 3x =

27.- 4 256x =

28.- 4 32x =

29.- 8 1024x =

30.- 116 4x− =

31.- 127 81x+ =

32.- 164 16x− =

33.- 1642

x =

34.- 1381

x =

35.- 2 1525

x =

36.- 8 2x− =


E - 5

37.- 1 162

x⎛ ⎞ =⎜ ⎟⎝ ⎠

38.- 11 279

x+⎛ ⎞ =⎜ ⎟⎝ ⎠

39.- 2 14 8x x−=

40.- 2 4 39 3x x− +=

41.- 2 2125 5x x− +=

42.- 2 44 2x x− +=


E - 6

7.3 Exponentials Base “e”

Use a scientific calculator to evaluate the following exponential expressions. Round your answers to two decimals.

1.- 2e

2.- 2.5

3e

3.- 4.5e

4.- 1 3e

5.- eπ

6.- 2e−

7.- 3 e

8.- 4eπ

9.- 2e π

10.- 2e π⋅

Applications of exponentials base “e” The population of aphids in a rose plant is given by the following formula:

0.1780 tP e ⋅= where “t” is the time (in weeks) since the plant is inspected. Find the aphid population at the following times after inspection:

11.- Now (0 weeks)

12.- After 1 week

13.- After 4 weeks

14.- After 12 weeks


E - 7

Use the continuously compounded interest formula rtA P e⎡ ⎤=⎣ ⎦ to find the total

accumulated in an account after the given number of years and at the specified annual interest rate. Round your answers to the cent.

Principal

Interest # years

15.- $1500

7% 1

16.- $2100

6.5% 2

17.- $1300

5.8% 2

18.- $1800

7.2% 2.5

19.- $2500

6.3% 3

20.- $3200

5.8% 3.5

21.- $1450

6.7% 2.5

22.- $3570

6.2% 2

Use the continuously compounded interest formula rtA P e⎡ ⎤=⎣ ⎦ to find the amount of

money [ ]P that should be invested at the given annual interest rate and for the specified number of years in order to accumulate the requested values. Round your answer to the cent.

# Interest # years Accumulated 23.- 7%

10 $50,000

24.- 6%

12 $35,000

25.- 6.6%

9 $40,000

26.- 5.8%

5 $25,000

27.- 7.1%

11 $70,000

28.- 6.5%

20 $100,000


E - 8

7.4 Logarithms

Write the following exponential equations in logarithmic form.

1.- 35 125=

2.- 1 24 2=

3.- 310 1000=

4.-

1 122

− =

5.- 2 139

− =

6.- 410 0.0001− =

7.- 1 3 36 6=

8.- 110 10=

9.- 2 16x =

10.- 3 3x =

11.- 0 1a =

12.- ye a=

13.- 1e e=

14.- 0 1e =

15.- ee y=

16.- 4e a− =

17.- 1e x− =

18.- xe y− =

19.- 2xe y=

20.- ye π=


E - 9

Write the following logarithmic equations in exponential form.

21.- ( )2log 8 3=

22.- ( )3log 81 4=

23.- ( )51log 52

=

24.- 71log 17

⎛ ⎞ = −⎜ ⎟⎝ ⎠

25.- ( )log 10 1=

26.- ( )ln 3 x=

27.- ( )log 1 0b =

28.- ( )2log 5x =

29.- ( )4log 2x = −

30.- ( ) 2log3

x = −

Without the use of a calculator evaluate the following logarithms (Hint: write the expression in exponential form)

31.- ( )6log 36 ?=

32.- ( )4log 64 ?=

33.- ( )3log 81 ?=

34.- ( )log 1,000,000 ?=

35.- 51log5

?⎛ ⎞ =⎜ ⎟⎝ ⎠

36.- ( )7log 7 ?=

37.- ( )32log 4 ?=

38.- ( )log 0.001 ?=


E - 10

Use a calculator to evaluate the following logarithms. Round your answer to five decimal places.

39.- ( )log 20

40.- ( )log 1.67

41.- ( )log 784

42.- ( )log 0.06

43.- 4log327

⎛ ⎞⎜ ⎟⎝ ⎠

44.- ( )log π

45.- ( )ln 2

46.- ( )ln 10

47.- ( )ln 58

48.- 1lnπ

⎛ ⎞⎜ ⎟⎝ ⎠

Use a calculator and the change of base formula to evaluate the following logarithms. Round your answers to five decimal places.

49.- ( )2log 100 ?=

50.- ( )3log 79 ?=

51.- ( )5log 0.4 ?=

52.- 36

1log100

?⎛ ⎞ =⎜ ⎟⎝ ⎠

53.- ( )1 4log 5 ?=

54.- ( )log 0.0 ?001π =

55.- ( )100log ?e =


E - 11

7.5 Properties of Logarithms

Use the properties of logarithms to write each expression in terms of the logarithms of individual variables or numbers.

1.- ( )log 5b x

2.- ( )log 2b xy

3.- ( )4logb y

4.- ( )2 5log 3b x

5.- ( )2 3logb x y z

6.- 5log

3bx⎛ ⎞

⎜ ⎟⎝ ⎠

7.-

logbxyz

⎛ ⎞⎜ ⎟⎝ ⎠

8.-

3logbxy

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

9.- 1log

2b⎛ ⎞⎜ ⎟⎝ ⎠

10.- 3

log6bx y⎛ ⎞

⎜ ⎟⎜ ⎟⎝ ⎠

11.-

31logb xy

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

12.- 2

3logbxyz

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

13.- ( )logb x

14.- ( )2logb x y


E - 12

15.- 2logbyx

⎛ ⎞⎜ ⎟⎝ ⎠

16.-

log4bxy

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

17.-

2logbxyz

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

18.-

log5bx⎛ ⎞

⎜ ⎟⎜ ⎟⎝ ⎠

19.-

3 2logbxy

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

20.-

log 2 byz

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

Given that ( )log 2 0.12b = , ( )log 3 0.19b = and ( )log 5 0.28b = , and using the properties of logarithms find the following logarithms:

21.- ( )log 6b

22.- ( )log 10b

23.- 3log

5b⎛ ⎞⎜ ⎟⎝ ⎠

24.- 15log

2b⎛ ⎞⎜ ⎟⎝ ⎠

25.- ( )log 9b

26.- ( )log 20b

27.- ( )log 5b

28.- 3log

2b⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠


E - 13

29.- ( )log 0.01b

30.- ( )log 0.003b

Use the properties of logarithms to write each combination of individual logarithms as a single logarithmic expression.

31.- ( ) ( )log 2 logb b x+

32.- ( ) ( )log 2 1 log 5b bx x− − −

33.- ( ) ( ) ( )log log logb b ba y x+ −

34.- ( ) ( ) ( )log log logb b bx y z− −

35.- ( ) ( )4log logb by z+

36.- ( ) ( )log 6 2logb b x+

37.- ( ) ( )3log 2logb bx y−

38.- ( ) ( )1 log log 32 b bx −

39.- ( ) ( )( )1 log log2 b ba y+

40.- ( ) ( )1 log 3log2 b ba z−

41.- ( ) ( ) ( )12log log 3log2b b bx y z+ −

42.- ( ) ( ) ( )2 log 2 2log log3 b b bx y− −

43.- ( ) ( ) ( )1 log 3log 3 2log3 b b ba x+ −

44.- ( ) ( )log 2 log 3b bx x− + +

45.- ( ) ( )log 1 log 1b bx x+ + −

46.- ( ) ( ) ( )log 3 2 log 2 1 logb b bx x x+ + − −


E - 14

7.6 Solving Exponential and Logarithmic Equations

Write as an equivalent logarithmic or exponential equation to solve for the unknown: (In case of an irrational answer, round it to 4 decimal places)

1.- 4 9x =

2.- 5xe =

3.- ( )ln 2x = −

4.- 13 2x− =

5.- ( )log 2.1x =

6.- ( )4

1log 12

x + =

7.- 2 12 5x+ =

8.- 210 0.18x =

9.- ( )2log 2 3x+ =

10.- ( )3log 2 1 4x− =

11.- ( )log 3 5 2x− =

12.- ( )ln 3 1.5x+ =

13.- ( )4log 3 2.7x− =

14.- 2 2 0.04xe + =


E - 15

In the following exercises use the uniqueness property of Logarithms to solve for x:

15.- ( ) ( )5 5log log 7x =

16.- ( ) ( )4 4log 2 log 8x + =

17.- ( ) ( )8 8log 6 log 2x− =

18.- ( ) ( )2 2log 5 log 3 1x x+ = −

19.- ( ) ( )3 3log 2 7 log 6 1x x+ = +

20.- ( ) ( )3 3log 2 1 log 2 1x x+ = −

21.- ( ) ( )27 7log 1 log 17x + =

22.- ( ) ( )25 5log 5 5 log 4 15x x x− + = −

23.- ( ) ( )26 6log 2 5 log 7x x x+ − = +

24.- ( ) ( )24 4log 2 logx x− =

25.- ( ) ( ) ( )5 5 5log log 12 log 4x x= − +

26.- ( ) ( ) ( )8 8 8log log 2 log 15x x+ − =

27.- ( ) ( ) ( )7 7 7log 3 log 6 2 log 3x x x− = − −

28.- ( ) ( ) ( )6 6 6log 5 log 1 log 19x x x− + + = +


E - 16

Answers to Problems on: 7.1 Inverse Functions # answer # answer # answer 1 Yes, it is. Inverse:

( ) ( ) ( ) ( ){ }1,3 , 2,4 , 0, 2 , 3,5− 5 ( )1 2f x x− = −

9 ( )1 2 6f x x− = −

2 Not one-to-one. Same y-coordinate 3 for x=2 and for x=-1

6 ( )1 6f x x− = +

10 ( )1 13

2f x x− = +

3 Yes, it is. Inverse:

( ) ( ) ( ) ( ){ }1,4 , 4,1 , 2,3 , 3,2 7

( )1 12xf x− −=

11 ( )1 2 1f x x− = +

4 Yes, it is. Inverse: ( ) ( ) ( )( )5,5 , 6, 4 , 0, 3 ,

4, 2

⎧ ⎫−⎪ ⎪⎨ ⎬− −⎪ ⎪⎩ ⎭

8

( )1 74xf x− +=

12

( )1 5 6f x x− = −

Alternative phrasing: # answer # answer # answer 5

2y x− = 8 7

4y x+ =

11 2 1y x+ =

6 6y x+ = 9 2 6x y= − 12 5 6y x− = 7 1

2y x− =

10 132

x y= +

Answers to Problems on: 7.2 Exponential Expressions and Equations # answer # answer # answer 1

81 15

83 29 10

3x =

2 13

16 33 y⋅

30 32

x =

3 1−

17 32 x⋅

31 13

x =

4 54

18

2 24 x− 32 5

3x =

5 13

19

4 82 x+ 33 1

6x = −

6 12

20

$2288.89 34

4x = −

7 2 21 $3898.69 35 1x = − 8 4

5

22 $4777.31

36 13

x = −

9 1 23 $2775.72 37 4x = − 10

25 24

$3301.58 38 5

2x = −

11 42 25 3x = 39 3x = −


E - 17

12 43

26 12

x = 40 11

3x =

13 62 27 4x = 41 4x = 14

34 28 5

2x =

42 8x =

Answers to Problems on: 7.3 Exponentials Base “e” # answer # answer # answer 1 7.39≈ 11 80= aphids 21 $1714.40≈ 2 4.06≈ 12 95≈ aphids 22 $4041.30≈ 3 59.65≈ 13 158≈ aphids 23 $24,829.27≈ 4 1.40≈ 14 615≈ aphids 24 $17,036.33≈ 5 23.14≈ 15 $1608.76≈ 25 $22,084.58≈ 6 0.14≈ 16 $2391.54≈ 26 $18,706.59≈ 7 4.95≈ 17 1459.89≈ 27 $32,056.35≈ 8 2.19≈ 18 $2154.99≈ 28 $27,253.18≈ 9 1.89≈ 19 $3020.10≈

10 535.49≈ 20 $3920.23≈ Answers to Problems on: 7.4 Logarithms # answer # answer # answer 1 ( )5log 125 3= 20 ( )ln yπ = 39 1.30103 2 ( )4

1log 22

= 21 32 8= 40 0.22272

3 ( )log 1000 3= 22 43 81= 41 2.89432 4

21log 12

⎛ ⎞ = −⎜ ⎟⎝ ⎠ 23 1 25 5=

42 -1.22185

5 31log 29

⎛ ⎞ = −⎜ ⎟⎝ ⎠ 24 1 17

7− = 43

-1.91249

6 ( )log 0.0001 4= − 25 110 10= 44 0.49715 7 ( )3

61log 63

= 26 3xe =

45 0.69315

8 ( )log 10 1= 27 0 1b = 46 2.30258 9 ( )2log 16 x= 28 52 x= 47 4.06044

10 ( )3log 3 x= 29 24 x− = 48 -1.14473

11 ( )log 1 0a = 30 2 310 x− = 49 6.64386≈ 12 ( )ln a y= 31 2 50 3.97724≈ 13 ( )ln 1e = 32 3 51 0.56932≈ − 14 ( )ln 1 0= 33 4 52 1.28510≈ −


E - 18

15 ( )ln y e= 34 6 53 1.16096≈ − 16 ( )ln 4a = − 35 1− 54 8.04586≈ − 17 ( )ln 1x = − 36 1

2 55 0.21715≈

18 ( )ln y x= − 37 23

19 ( )ln 2y x= 38 3− Answers to Problems on: 7.5 Properties of Logarithms

# answer # answer # answer 1

( ) ( )log 5 logb b x+

17 ( ) ( )( )

1 1log log2 22log

b b

b

x y

z

+

−

33

logbayx

⎛ ⎞⎜ ⎟⎝ ⎠

2 ( ) ( ) ( )log 2 log logb b bx y+ +

18 ( ) ( )1 1log log 5

2 2b bx − 34

logbxyz

⎛ ⎞⎜ ⎟⎝ ⎠

3 ( )4logb y

19 ( ) ( )1 2log log

3 3b bx y− 35 ( )4logb y z

4

( ) ( )2log 3 5logb b x+

20 ( ) ( )

( )

1log 2 log2

1 log2

b b

b

y

z

+

−

36

( )2log 6b x

5

( ) ( ) ( )2log 3log logb b bx y z+ + 21

0.31 37 3

2logbxy

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

6

( ) ( ) ( )log 5 log log 3b b bx+ − 22

0.40 38

log3bx⎛ ⎞

⎜ ⎟⎜ ⎟⎝ ⎠

7 ( ) ( ) ( )log log logb b bx y z− − 23 0.09− 39 ( )logb ay

8 ( ) ( )log 3logb bx y−

24 0.35

40

3logbaz

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

9 ( )log 2b−

25 0.38

41 2

3logbx yz

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

10

( ) ( ) ( )3log log log 6b b bx y+ − 26

0.52 42 3

24logb x y

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

11 ( ) ( )log 3logb bx y− −

27 0.14

43 3

227logb

ax

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

12 ( ) ( ) ( )log 2log 3logb b bx y z+ −

28 0.025−

44 ( )2log 6b x x+ −

13 ( )1 log

2 b x 29

0.8− 45 ( )2log 1b x −


E - 19

14

( ) ( )12log log2b bx y+

30 1.01−

46 26 2logbx xx

⎛ ⎞+ −⎜ ⎟⎜ ⎟⎝ ⎠

15 ( ) ( ) ( )1log 2 log log

2b b by x+ − 31

( )log 2b x

16 ( ) ( ) ( )1 log log 4 log

2 b b bx y− −

32 2 1log5b

xx−⎛ ⎞

⎜ ⎟−⎝ ⎠

Answers to Problems on: 7.6 Solving Exponential and Logarithmic Equations # answer # answer # answer 1

1.5850x ≈ 11

35x = 21 4x = ± (both check –

so both are solutions) 2

1.6094x ≈ 12

1.4817x ≈ 22 4; 5x x= = (both

check – so both are solutions)

3

0.1353x ≈ 13

39.2243x ≈ − 23 4; 3x x= − = (both

check – so both are solutions)

4

1.6309x ≈

14

2.6094x ≈ −

24 2; 1x x= = − (X = -1 does not check – so x = 2 is the only solution)

5

125.8925x ≈

15 7x = (checks – so it

is a solution)

25 6 ; 2x x= − = (x = -6 does not check – so x = 2 is the only solution)

6

1x =


is a solution)

26 5 ; 3x x= = − (x = -3 does not check – so x = 5 is the only solution)

7

0.6610x ≈


is a solution)

27 2 ; 33

x x= − =

(neither check – so no solution)

8

0.3724x ≈ −


is a solution)

28 8 ; 3x x= = − (x = -3 does not check – so x = 8 is the only solution)

9

6x = 19 3

2x = (checks – so it

is a solution)

10

41x = 20 0 2≠ −

Absurd answer à No solution

dgriffinresources.netdgriffinresources.net/math102ch7book.pdf · 7% 1 16.- $2100 6.5% 2 17.- $1300...

Documents