Deck 21: Exponential and Logarithmic Equations

Question 1

Solve the exponential equation algebraically.Approximate the result to three decimal places.​ $3 ^ { 8 x } = 2400$ ​

A) $\frac { \ln 2400 } { 8 \ln 3 } \approx 0.018$
B) $\frac { \ln 2400 } { 8 \ln 3 } \approx 0.886$
C) $\frac { \ln 2400 } { 8 \ln 3 } \approx 0.089$
D) $\frac { \ln 2400 } { 8 \ln 3 } \approx - 0.886$
E) $\frac { \ln 2400 } { 8 \ln 3 } \approx 1.248$

Answer 1

$\frac { \ln 2400 } { 8 \ln 3 } \approx 0.886$

Question 2

Determine whether the given x-value is a solution (or an approximate solution)of the equation.​ $\begin{array} { l } 4 ^ { 3 x - 10 } = 16 \\x = 4\end{array}$ ​

A)No
B)Yes

Answer 2

Yes

Question 3

Solve the exponential equation algebraically.Approximate the result to three decimal places.​ $2 ^ { x - 5 } = 16$ ​

A)10
B)11
C)9
D)-9
E)12

Answer 3

9

Question 4

Solve for x.​ $\ln x - \ln 5 = 0$ ​

A) $$- \frac { 1 } { 5 }$$
B) $- 5$
C) $\frac { 1 } { 5 }$
D) $5 ^ { - 2 }$
E)5

Question 5

Solve the exponential equation algebraically.Approximate the result to three decimal places.​ $3 e ^ { x } = 15$ ​

A) $\ln 5 \approx - 1.609$
B) $\ln 5 \approx 1.099$
C) $\ln 5 \approx - 1.099$
D) $\ln 5 \approx 1.609$
E) $\ln 5 \approx 2.708$

Question 6

Solve for x.Approximate the result to three decimal places.​ $e ^ { x } = 4$ ​

A) $\ln 4 \approx 1.466$
B) $\ln 4 \approx 1.476$
C) $\ln 4 \approx 1.436$
D) $\ln 4 \approx 1.386$
E) $\ln 4 \approx 1.456$

Question 7

Solve the exponential equation algebraically.Approximate the result to three decimal places.​ $2 \left( 3 ^ { x } \right) = 36$ ​

A) $\frac { \ln 18 } { \ln 3 } \approx 0.380$
B) $\frac { \ln 18 } { \ln 3 } \approx - 0.380$
C) $\frac { \ln 18 } { \ln 3 } \approx - 2.631$
D) $\frac { \ln 3 } { \ln 18 } \approx 2.631$
E) $\frac { \ln 18 } { \ln 3 } \approx 2.631$

Question 8

Solve the exponential equation algebraically.Approximate the result to three decimal places.​ $e ^ { x } - 6 = 13$ ​

A) $\ln 19 \approx - 2.565$
B) $\ln 19 \approx 2.944$
C) $\ln 19 \approx 1.792$
D) $\ln 19 \approx 2.565$
E) $\ln 19 \approx - 2.944$

Question 9

Solve for x.​ $\left( \frac { 1 } { 4 } \right) ^ { x } = 64$ ​

A)4
B) $$- 3$$
C)7
D)3
E) $- 4$

Question 10

Solve for x.Approximate the result to three decimal places.​ $\ln x = - 2$ ​

A) $e ^ { - 2 } \approx 7.389$
B) $e ^ { - 2 } \approx 0.368$
C) $e ^ { - 2 } \approx 2.718$
D) $e ^ { - 2 } \approx 0.135$
E) $e ^ { - 2 } \approx 1.000$

Question 11

Solve for x.​ $\log x = - 4$ ​

A) $- 4$
B)0.0001
C) $- 0.0001$
D) $4$
E)1.386

Question 12

Determine whether the given x-value is a solution (or an approximate solution)of the equation.​ $\begin{array} { l } 4 ^ { 5 x - 5 } = 256 \\x = 2\end{array}$ ​

A)No
B)Yes

Question 13

Solve the exponential equation algebraically.Approximate the result to three decimal places.​ $4 e ^ { x } = 33$ ​

A) $\ln \frac { 33 } { 4 } \approx 2.110$
B) $\ln \frac { 33 } { 4 } \approx - 2.110$
C) $\ln \frac { 4 } { 33 } \approx 2.110$
D) $\ln \frac { 33 } { 4 } \approx 3.497$
E) $\ln \frac { 33 } { 4 } \approx 0.114$

Question 14

Solve the exponential equation algebraically.Approximate the result to three decimal places.​ $e ^ { - 4 x ^ { 2 } } = e ^ { 3 x ^ { 2 } - 14 x }$ ​

A) $4,2$
B) $3,2$
C) $- 4,2$
D) $0,2$
E) $0 , - 2$

Question 15

Solve for x.​ $6 ^ { x } = 36$ ​

A)8
B)2
C)-2
D)-6
E)6

Question 16

Solve the exponential equation algebraically.Approximate the result to three decimal places.​ $e ^ { x } = e ^ { x ^ { 2 } - 12 }$ ​

A) $3 , - 4$
B) $- 3 , - 4$
C) $3,4$
D) $- 3,4$
E) $- 3,3$

Question 17

Solve the exponential equation algebraically.Approximate the result to three decimal places.​ $6 ^ { x } + 8 = 57$ ​

A) $\frac { \ln 49 } { \ln 6 } \approx 2.172$
B) $\frac { \ln 49 } { \ln 6 } \approx - 0.460$
C) $\frac { \ln 49 } { \ln 6 } \approx 0.514$
D) $\frac { \ln 49 } { \ln 6 } \approx 0.460$
E) $\frac { \ln 49 } { \ln 6 } \approx - 2.172$

Question 18

Solve for x.Approximate the result to three decimal places.​ $\log _ { 7 } x = \frac { 1 } { 2 }$ ​

A) $\sqrt { 7 } \approx 2.828$
B) $\sqrt { 7 } = 2.646$
C) $\sqrt { 7 } \approx - 2.646$
D) $$\sqrt { 7 } \approx 2.646$$
E) $\frac { \sqrt { 7 } } { 7 } \approx 2.646$

Question 19

Solve the exponential equation algebraically.Approximate the result to three decimal places.​ $6 ^ { - 5 x } = 0.50$ ​

A) $- \frac { \ln ( 0.50 ) } { 5 \ln 6 } \approx 0.072$
B) $\frac { \ln ( 0.50 ) } { 5 \ln 6 } \approx 0.517$
C) $- \frac { \ln ( 0.50 ) } { 5 \ln 6 } \approx 0.387$
D) $- \frac { \ln ( 0.50 ) } { 5 \ln 6 } \approx 0.077$
E) $- \frac { \ln ( 0.50 ) } { 5 \ln 6 } \approx - 0.077$

Question 20

Solve the exponential equation algebraically.Approximate the result to three decimal places.​ $e ^ { x } = e ^ { x ^ { 2 } - 30 }$ ​

A) $- 6 , - 5$
B) $1 , - 30$
C) $6,5$
D) $- 6,5$
E) $6 , - 5$

Question 21

Solve the logarithmic equation algebraically.Approximate the result to three decimal places.​ $9 + 2 \ln x = 4$ ​

A) $e ^ { - 5 / 2 } \approx 1.082$
B) $e ^ { - 5 / 2 } \approx - 0.918$
C) $e ^ { - 5 / 2 } \approx - 1.918$
D) $e ^ { - 5 / 2 } \approx 0.082$
E) $e ^ { - 5 / 2 } \approx 2.082$

Question 22

$2500 is invested in an account at interest rate r,compounded continuously.Find the time required for the amount to double.(Approximate the result to two decimal places. )​ $r = 0.06$ ​

A)13.55 yr
B)10.55 yr
C)12.55 yr
D)9.55 yr
E)11.55 yr

Question 23

Solve the logarithmic equation algebraically.Approximate the result to three decimal places.​ $\ln x = - 5$ ​

A) $e ^ { - 5 } \approx 2.007$
B) $e ^ { - 5 } \approx 1.007$
C) $e ^ { - 5 } \approx 3.007$
D) $e ^ { - 5 } \approx - 1.993$
E) $e ^ { - 5 } \approx 0.007$

Question 24

Solve the exponential equation algebraically.Approximate the result to three decimal places.​ $2 \left( 5 ^ { 2 - x } \right) = 12$ ​

A) $2 - \frac { \ln 6 } { \ln 5 } \approx 3.887$
B) $2 - \frac { \ln 6 } { \ln 5 } \approx 1.887$
C) $2 - \frac { \ln 6 } { \ln 5 } \approx 0.887$
D) $2 - \frac { \ln 6 } { \ln 5 } \approx 4.887$
E) $2 - \frac { \ln 6 } { \ln 5 } \approx 2.887$

Question 25

Solve the logarithmic equation algebraically.Approximate the result to three decimal places.​ $\ln \sqrt { x - 8 } = 7$ ​

A) $e ^ { 14 } + 8 \approx 1,202,611.284$
B) $e ^ { 14 } + 8 \approx 1,202,614.284$
C) $e ^ { 14 } + 8 \approx 1,202,612.284$
D) $e ^ { 14 } + 8 \approx 1,202,610.284$
E) $e ^ { 14 } + 8 \approx 1,202,613.284$

Question 26

$3000 is invested in an account at interest rate r,compounded continuously.Find the time required for the amount to triple.(Approximate the result to two decimal places. )​ $r = 0.03$ ​

A)37.62 yr
B)36.62 yr
C)35.62 yr
D)34.62 yr
E)38.62 yr

Question 27

Solve the exponential equation algebraically.Approximate the result to three decimal places.​ $- 22 + 4 e ^ { x } = 13$ ​

A) $\ln \frac { 35 } { 4 } \approx 1.169$
B) $\ln \frac { 35 } { 4 } \approx 4.169$
C) $\ln \frac { 35 } { 4 } \approx 2.169$
D) $\ln \frac { 35 } { 4 } \approx 0.169$
E) $\ln \frac { 35 } { 4 } \approx 3.169$

Question 28

Solve the exponential equation algebraically.Approximate the result to three decimal places.​ $e ^ { 3 x } = 75$ ​

A) $\frac { \ln 75 } { 3 } \approx 0.439$
B) $\frac { \ln 75 } { 3 } \approx 2.439$
C) $\frac { \ln 75 } { 3 } \approx - 0.561$
D) $\frac { \ln 75 } { 3 } \approx 3.439$
E) $\frac { \ln 75 } { 3 } \approx 1.439$

Question 29

Solve the exponential equation algebraically.Approximate the result to three decimal places.​ $4 \left( 5 ^ { x - 5 } \right) = 24$ ​

A) $5 + \frac { \ln 6 } { \ln 5 } \approx 7.113$
B) $5 + \frac { \ln 6 } { \ln 5 } \approx 9.113$
C) $5 + \frac { \ln 6 } { \ln 5 } \approx 6.113$
D) $5 + \frac { \ln 6 } { \ln 5 } \approx 8.113$
E) $5 + \frac { \ln 6 } { \ln 5 } \approx 10.113$

Question 30

Solve the logarithmic equation algebraically.Approximate the result to three decimal places.​ $8 + 7 \ln x = 14$ ​

A) $e ^ { 6 / 7 } \approx 3.356$
B) $e ^ { 6 / 7 } \approx 1.356$
C) $e ^ { 6 / 7 } \approx 4.356$
D) $e ^ { 6 / 7 } \approx 2.356$
E) $e ^ { 6 / 7 } \approx 0.356$

Question 31

Solve the logarithmic equation algebraically.Approximate the result to three decimal places.​ $\ln 2 x = 3$ ​

A) $\frac { e ^ { 3 } } { 2 } \approx 8.043$
B) $\frac { e ^ { 3 } } { 2 } \approx 10.043$
C) $\frac { e ^ { 3 } } { 2 } \approx 12.043$
D) $\frac { e ^ { 3 } } { 2 } \approx 11.043$
E) $\frac { e ^ { 3 } } { 2 } \approx 13.043$

Question 32

Solve the logarithmic equation algebraically.Approximate the result to three decimal places.​ $2 \log _ { 3 } ( 0.5 x ) = 5$ ​

A) $2 \left( 3 ^ { 5 / 2 } \right) \approx 33.177$
B) $2 \left( 3 ^ { 5 / 2 } \right) \approx 30.177$
C) $2 \left( 3 ^ { 5 / 2 } \right) \approx 31.177$
D) $2 \left( 3 ^ { 5 / 2 } \right) \approx 29.177$
E) $2 \left( 3 ^ { 5 / 2 } \right) \approx 32.177$

Question 33

Solve the logarithmic equation algebraically.Approximate the result to three decimal places.​ $\ln x - 5 = 0$ ​

A) $e ^ { 5 } \approx 151.413$
B) $e ^ { 5 } \approx 146.413$
C) $e ^ { 5 } \approx 148.413$
D) $e ^ { 5 } \approx 149.413$
E) $e ^ { 5 } \approx 150.413$

Question 34

Solve the exponential equation algebraically.Approximate the result to three decimal places.​ $e ^ { 2 x } - 7 e ^ { x } - 8 = 0$ ​

A) $\ln 8 \approx 3.079$
B) $\ln 8 \approx 2.079$
C) $\ln 8 \approx 4.079$
D) $\ln 8 \approx 1.079$
E) $\ln 8 \approx 0.079$

Question 35

Solve the exponential equation algebraically.Approximate the result to three decimal places.​ $e ^ { 2 x } - 7 e ^ { x } - 8 = 0$ ​

A) $\ln 8 \approx 3.079$
B) $\ln 8 \approx 1.079$
C) $\ln 8 \approx 2.079$
D) $\ln 8 \approx 0.079$
E) $\ln 8 \approx 4.079$

Question 36

Solve the exponential equation algebraically.Approximate the result to three decimal places.​ $1000 e ^ { - 4 x } = 75$ ​

A) $- \frac { 1 } { 4 } \ln \frac { 3 } { 40 } \approx 2.648$
B) $- \frac { 1 } { 4 } \ln \frac { 3 } { 40 } \approx - 1.352$
C) $- \frac { 1 } { 4 } \ln \frac { 3 } { 40 } \approx 1.648$
D) $- \frac { 1 } { 4 } \ln \frac { 3 } { 40 } \approx 0.648$
E) $- \frac { 1 } { 4 } \ln \frac { 3 } { 40 } \approx - 0.352$

Question 37

Solve the logarithmic equation algebraically.Approximate the result to three decimal places.​ $6 \ln x = 5$ ​

A) $e ^ { 5 / 6 } \approx 0.301$
B) $e ^ { 5 / 6 } \approx 1.301$
C) $e ^ { 5 / 6 } \approx 3.301$
D) $e ^ { 5 / 6 } \approx 2.301$
E) $e ^ { 5 / 6 } \approx 4.301$

Question 38

$5500 is invested in an account at interest rate r,compounded continuously.Find the time required for the amount to double.(Approximate the result to two decimal places. )​ $r = 0.0540$ ​

A)13.84 yr
B)10.84 yr
C)12.84 yr
D)11.84 yr
E)14.84 yr

Question 39

Solve the logarithmic equation algebraically.Approximate the result to three decimal places.​ $\log x = 4$ ​

A)1,000
B)1,000,000
C)10,000
D)100,000
E)100

Question 40

Solve the logarithmic equation algebraically.Approximate the result to three decimal places.​ $\log _ { 5 } ( 5 x - 3 ) = \log _ { 5 } ( 4 x + 4 )$ ​

A)8
B)6
C)-1
D)7
E)9

Question 41

Solve for x: $7 \left( 10 ^ { x - 3 } \right) = 23$ .Round to 3 decimal places.

A)0.517
B)no solution
C)3.517
D)-1.362
E)1.362

Question 42

Solve for x: $5 ^ { - x / 3 } = 0.0052$ .Round to 3 decimal places.

A)-20.606
B)9.803
C)20.606
D)-3.268
E)15.777

Question 43

Solve the equation $\log ( 1 - x ) = \log ( 10 )$ for x using the One-to-One Property.

A)11
B)-9
C)-11
D)0
E)The equation has no solution.

Question 44

Solve the equation algebraically.Round the result to three decimal places.​ $\frac { 3 + \ln x } { 8 } = 0$ ​

A) $e ^ { - 3 } \approx 2,980.958$
B) $e ^ { - 3 } \approx 0$
C) $e ^ { - 3 } \approx 54.598$
D) $e ^ { - 3 } \approx 0.050$
E) $e ^ { - 3 } \approx 20.086$

Question 45

Simplify the expression.​ $\log _ { 5 } 5 ^ { 9 }$ ​

A)45
B)5
C)9
D)81
E)none of these

Question 46

Solve $\left( \frac { 1 } { 4 } \right) ^ { x } = 64$ for x.

A)-3
B)-4
C)-1
D)1
E)no solution

Question 47

Determine whether or not $x = \frac { 3 } { 7 }$ is a solution to $3 ^ { 3 x - 3 } = 81$ .

A)no
B)yes

Question 48

Use the One-to-One Property to solve the following equation for x. $2 ^ { 5 x } = 128$

A) $\frac { 128 } { 5 }$
B) $- \frac { 64 } { 5 }$
C) $\frac { 7 } { 5 }$
D) $\frac { 5 } { 7 }$
E)2

Question 49

Solve the equation.Round to 3 decimal places. ​
5^x = 8
​

A)x = 1.415
B)x = 2.292
C)x = 6.724
D)x = 2.526
E)x = 1.292

Question 50

Simplify the expression.​ $6 ^ { \log _ { 6 } 4 }$ ​

A)4
B)16
C)24
D)6
E)none of these

Question 51

$4500 is invested in an account at interest rate r,compounded continuously.Find the time required for the amount to triple.(Approximate the result to two decimal places. )​ $r = 0.0315$ ​

A)21.00 yr
B)20.00 yr
C)34.88 yr
D)24.00 yr
E)23.00 yr