Exponential and Logarithmic Functions: Interactive Quizzes and Practice Questions

Question 1

Which graph represents the function f(x) = 2^x?

Accepted Answer

An exponential curve that increases rapidly

Answer

A straight line with a positive slope

Answer

A straight line with a negative slope

Answer

An exponential curve that decreases rapidly

Question 2

What is the definition of a logarithmic function?

Accepted Answer

A function that is the inverse of exponential functions

Answer

A function that reverses the operation of exponential functions

Answer

A function that performs the same operation as exponential functions

Answer

A function that is the opposite of exponential functions

Question 3

Which of the following is equivalent to e^(ln(2x))?

Accepted Answer

2x

Answer

ln(2x)

Answer

2

Answer

1

Question 4

Solve the logarithmic equation: log(x + 5) = 2

Accepted Answer

x = 7

Answer

x = 1

Answer

x = 5

Answer

x = 9

Question 5

What is the inverse function of f(x) = e^x?

Accepted Answer

f^-1(x) = log(x)

Answer

f^-1(x) = -log(x)

Answer

f^-1(x) = sqrt(x)

Answer

f^-1(x) = x^2

Question 6

Find the domain of the function f(x) = log(x^2 - 4).

Accepted Answer

x > 2 or x < -2

Answer

x > 2 and x < -2

Answer

x > 0

Answer

All real numbers

Question 7

Which of the following is the equation of the exponential function with base b and exponent x?

Accepted Answer

y = b^x

Answer

y = x^b

Question 8

What is the domain of the exponential function y = e^x?

Accepted Answer

All real numbers

Answer

All positive real numbers

Question 9

What is the range of the exponential function y = e^x?

Accepted Answer

All positive real numbers

Question 10

What is the inverse function of the exponential function y = 2^x?

Accepted Answer

y = log_2(x)

Answer

y = 2^(log_2(x))

Question 11

Solve the equation 2^x = 16:

Accepted Answer

x = 4

Question 12

Which of the following is the equation of the logarithmic function with base b?

Accepted Answer

y = log_b(x)

Answer

y = x^b

Question 13

The inverse of an exponential function is a function of the form:

Accepted Answer

y = log a(x), where a is the base of the exponential function.

Answer

y = x^a, where a is the base of the exponential function.

Answer

y = a^x, where a is the base of the exponential function.

Answer

y = log x(a), where a is the base of the exponential function.

Question 14

The expression log(a) - log(b) is equivalent to:

Accepted Answer

log(a/b)

Answer

log(a + b)

Answer

log(a - b)

Answer

log(b/a)

Question 15

Which of the following is the domain of the function f(x) = log(x - 2)?

Accepted Answer

x > 2

Answer

x ≥ 2

Answer

x < 2

Answer

x ≤ 2

Question 16

A population of bacteria grows exponentially with a doubling time of 3 hours. If the initial population is 1000, what is the population after 12 hours?

Accepted Answer

16000

Answer

2000

Answer

4000

Answer

8000

Question 17

Which of the following equations represents exponential growth? (Equation format: y = f(x))

Accepted Answer

y = 2^x

Answer

y = 2x - 3

Answer

y = log2x

Answer

y = 1/2x

Question 18

If a population grows at a rate of 5% per year, which of the following functions models the population growth? (Equation format: f(x) = population)

Accepted Answer

f(x) = 1.05^x

Answer

f(x) = 5x

Answer

f(x) = 5^x

Answer

f(x) = 0.05x

Question 19

Solve for y: 2^y = 16

Accepted Answer

y = 4

Answer

y = 2

Answer

y = 8

Answer

y = 16

Question 20

Which of the following is the range of the function g(x) = log2(x - 1)?

Accepted Answer

(1, ∞)

Answer

(-∞, 0)

Answer

(0, ∞)

Answer

[0, ∞)

Question 21

Which of the following is the inverse of the function f(x) = 2^x?

Accepted Answer

f^-1(x) = log2x

Answer

f^-1(x) = log10x

Answer

f^-1(x) = x^2

Answer

f^-1(x) = 2x

Question 22

What are the domains and ranges of exponential and logarithmic functions?

Accepted Answer

Exponential functions: all real numbers (domain) and positive numbers (range)

Accepted Answer

Logarithmic functions: positive numbers (domain) and all real numbers (range)

Answer

Exponential and logarithmic functions: all real numbers (both domain and range)

Answer

Domain and range limitations have no impact on the validity of equations with exponential and logarithmic functions.

Question 23

Which graph represents the exponential function (y = 2^x)?

Accepted Answer

Image of an upward-sloping curve

Answer

Image of a horizontal line passing through (0, 1)

Answer

Image of a downward-sloping curve

Question 24

The logarithmic function (log_2(x)) is the inverse of which exponential function?

Accepted Answer

(y = 2^x)

Answer

(y = 1/x)

Answer

(y = x^2)

Answer

(y = e^x)

Question 25

Evaluate (log_{10}(100)).

Accepted Answer

2

Answer

1

Answer

10

Answer

100

Question 26

Which property of logarithms is used to simplify (log_a(x) + log_a(y))?

Accepted Answer

(log_a(xy))

Answer

(log_a(x^y))

Answer

(log_a(x - y))

Answer

(log_a(x/y))

Question 27

A population grows exponentially according to the equation (P(t) = 100e^(0.05t)), where t is time in years. What is the population after 10 years?

Accepted Answer

223.13

Answer

105

Answer

1000

Answer

500

Question 28

Which graph uses a logarithmic scale?

Accepted Answer

Image of a graph with evenly spaced lines representing orders of magnitude

Answer

Image of a graph with evenly spaced lines representing absolute values

Answer

Image of a graph with non-uniformly spaced lines

Question 29

Identify the graph of the function y = log(x - 1).

Accepted Answer

Image of a graph showing a logarithmic function shifted 1 unit to the right.

Answer

Image of a graph showing an exponential function shifted 1 unit to the right.

Answer

Image of a graph showing a logarithmic function shifted 1 unit to the left.

Question 30

Identify an example of an exponential growth model.

Accepted Answer

The population of bacteria doubles every hour.

Answer

The temperature of a cup of coffee decreases by 5 degrees every minute.

Answer

The amount of money in a savings account increases by a fixed amount each year.

Answer

The height of a ball thrown in the air decreases by 10% each second.

Question 31

Use exponential regression to find the equation of the curve that best fits the following data: (1, 2), (2, 4), (3, 8), (4, 16).

Accepted Answer

y = 2^x

Answer

y = 4^x

Answer

y = 2x

Answer

y = x^2

Question 32

The population of a city is growing exponentially. In 2010, the population was 100,000. In 2020, the population was 150,000. What is the approximate annual growth rate of the population?

Accepted Answer

4.1%

Answer

3.3%

Answer

5.0%

Answer

2.5%

Question 33

The function f(x) = 2^(x+1) is shifted 3 units to the left. What is the equation of the shifted function?

Accepted Answer

f(x) = 2^(x+4)

Answer

f(x) = 2^(x-2)

Answer

f(x) = 2^(x-3)

Answer

f(x) = 2^(x+3)

Question 34

What is the domain of the function f(x) = log₂(x-3)?

Accepted Answer

x > 3

Answer

x < 3

Answer

All real numbers

Answer

x ≥ 3

Question 35

Solve for x: log₃(x+1) = 2

Accepted Answer

x = 8

Answer

x = 9

Answer

x = 2

Answer

x = 5

Question 36

Which of the following is equivalent to log₅(25)?

Accepted Answer

2

Answer

5

Answer

1/5

Answer

125

Question 37

The half-life of a radioactive substance is 10 years. If you start with 100 grams of the substance, how many grams will remain after 30 years?

Accepted Answer

12.5 grams

Answer

75 grams

Answer

25 grams

Answer

50 grams

Question 38

Which of the following is the inverse function of f(x) = e^x?

Accepted Answer

f⁻¹(x) = ln(x)

Answer

f⁻¹(x) = x^e

Answer

f⁻¹(x) = 1/e^x

Answer

f⁻¹(x) = e^(-x)

Question 39

The graph of the function y = log₂(x) is reflected across the y-axis. What is the equation of the reflected graph?

Accepted Answer

y = log₂(-x)

Answer

y = -log₂(x)

Answer

y = log₂(x+1)

Answer

y = log₂(1/x)

Question 40

The pH of a solution is given by the formula pH = -log[H+], where [H+] is the hydrogen ion concentration. If the pH of a solution is 4, what is the hydrogen ion concentration?

Accepted Answer

10⁻⁴

Answer

0.0001

Answer

4

Answer

10⁴

Question 41

Simplify the expression: log₃(27) + log₃(9) - log₃(3)

Accepted Answer

5

Answer

7

Answer

9

Answer

3

Question 42

The population of a city is growing exponentially. In 2010, the population was 100,000, and it is expected to double every 15 years. Which equation represents the population, P, t years after 2010?

Accepted Answer

P = 100000 * 2^(t/15)

Answer

P = 100000 + 2^(t/15)

Answer

P = 100000 * (1 + 1/15)^t

Answer

P = 100000 * (1 + 2)^t

Question 43

What is the domain of the function f(x) = log₂(x - 3)?

Accepted Answer

x > 3

Answer

x < 3

Answer

x ≥ 3

Answer

All real numbers

Question 44

Simplify the expression: log₃(27) + log₃(9)

Accepted Answer

5

Answer

3

Answer

6

Question 45

Solve for x: 2^(x+1) = 8

Accepted Answer

x = 2

Answer

x = -1

Answer

x = 1

Answer

x = 3

Question 46

Which of the following is equivalent to log₅(25)?

Accepted Answer

2

Answer

1/2

Answer

125

Answer

5

Question 47

The half-life of a radioactive substance is 10 years. If you start with 100 grams of the substance, how much will be left after 30 years?

Accepted Answer

12.5 grams

Answer

37.5 grams

Answer

50 grams

Answer

25 grams

Question 48

What is the inverse function of f(x) = e^(2x) ?

Accepted Answer

f⁻¹(x) = (ln(x))/2

Answer

f⁻¹(x) = e^(-2x)

Answer

f⁻¹(x) = ln(2x)

Answer

f⁻¹(x) = 2ln(x)

Question 49

Solve for x: log₄(x) + log₄(x - 3) = 1

Accepted Answer

x = 4

Answer

x = 2

Answer

x = 1

Answer

x = 3

Question 50

The graph of y = log₂(x) is reflected across the y-axis. What is the equation of the reflected graph?

Accepted Answer

y = log₂(-x)

Answer

y = -log₂(x)

Answer

y = log₂(x - 1)

Answer

y = log₂(x + 1)

Question 51

A scientist is studying the growth of bacteria in a petri dish. The initial population is 100 bacteria, and the population doubles every 20 minutes. Which equation represents the population, P, after t minutes?

Accepted Answer

P = 100 * 2^(t/20)

Answer

P = 100 + 2^(t/20)

Answer

P = 100 * (1 + 2)^t