Exponents and Logarithms: Master the Laws and Solve Equations

Question 1

Solve for x: log base 5 (x + 2) = 2

Accepted Answer

7

Answer

3

Answer

5

Answer

9

Question 2

Which of the following functions is the inverse of y = 2^x?

Accepted Answer

y = log base 2 (x)

Answer

y = e^x

Answer

y = 10^x

Answer

y = x^2

Question 3

A certain bacteria population triples every hour. If the initial population is 100, what is the population after 3 hours?

Accepted Answer

900

Answer

300

Answer

600

Answer

1200

Question 4

Simplify the expression: (3^2)^3

Accepted Answer

3^6

Answer

6^3

Answer

9^3

Answer

27^3

Question 5

Which of the following is NOT a property of logarithms?

Accepted Answer

loga^b = b^a

Answer

log(ab) = log(a) + log(b)

Answer

log(1) = 0

Answer

log(a/b) = log(a) - log(b)

Question 6

Which of the following graphs represents the function f(x) = 2^x?

Accepted Answer

An exponential curve with base 2

Answer

A straight line with positive slope

Answer

A straight line with negative slope

Answer

A logarithmic curve with base 2

Question 7

Find the inverse of the function f(x) = 3^x

Accepted Answer

f^-1(x) = log3(x)

Answer

f^-1(x) = x^3

Answer

f^-1(x) = 1/3^x

Answer

f^-1(x) does not exist

Question 8

Simplify the logarithmic expression: log(x * y^2)

Accepted Answer

log(x) + 2log(y)

Answer

log(x) + log(y)

Answer

log(x²) + log(y)

Answer

2log(xy)

Question 9

Simplify the following expression: (2^3)^4

Accepted Answer

2^12

Answer

2^7

Answer

4^7

Answer

4^12

Question 10

Find the value of x: 2x * 3^x = 54

Accepted Answer

4

Answer

2

Answer

3

Answer

6

Question 11

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

Accepted Answer

y = log2(x)

Answer

y = x^2

Answer

y = e^x

Answer

y = 10^x

Question 12

Apply the quotient rule of exponents to simplify the expression: x^5 / x^2

Accepted Answer

x^3

Answer

x^7

Answer

x^2

Question 13

Determine the value of log10(100)

Accepted Answer

2

Answer

1

Answer

3

Question 14

Which property of logarithms is demonstrated by the equation log(a^b) = bloga?

Accepted Answer

log(a^b) = bloga

Answer

loga - logb = log(a - b)

Answer

log(ab) = log a + log b

Answer

log(a + b) = loga + logb

Question 15

Simplify the expression: (3^2)^3

Accepted Answer

3^6

Answer

3^5

Answer

6^3

Answer

9^3

Question 16

Express 64 as a power of 2

Accepted Answer

2^6

Answer

2^3

Answer

2^4

Answer

2^5

Question 17

Simplify the expression: (2^4)^-2

Accepted Answer

2^-8

Answer

2^8

Answer

2^2

Answer

2^-2

Question 18

Simplify the expression (2^3)^2 using the laws of exponents.

Accepted Answer

2^6

Answer

3^6

Answer

6^2

Answer

2^5

Question 19

Express 4 as a power of 2.

Accepted Answer

2^2

Answer

2^-2

Answer

4^2

Answer

1/2^2

Question 20

Simplify the logarithmic expression log(x) + log(y) - log(z).

Accepted Answer

log(xy/z)

Answer

log(x + y - z)

Answer

log(xyz)

Answer

log(x) + log(y) - log(z)

Question 21

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

Accepted Answer

(2, ∞)

Answer

(-∞, 2)

Answer

(0, 2)

Answer

(-2, ∞)

Question 22

Which of the following is equivalent to the expression logb(m/n)?

Accepted Answer

logb(m) - logb(n)

Answer

logb(m + n)

Answer

logb(mn)

Answer

1/logb(mn)

Question 23

Which of the following is NOT a property of logarithms?

Accepted Answer

log(a + b) = log(a) + log(b)

Answer

log(ab) = log(a) + log(b)

Answer

log(1) = 0

Answer

log(a^b) = b log(a)

Question 24

Identify the inverse operation of taking the logarithm:

Accepted Answer

Exponentiation

Answer

Multiplication

Answer

Square root

Answer

Division

Question 25

Express log(ab) in terms of log a and log b:

Accepted Answer

log a + log b

Answer

log a - log b

Answer

log b - log a

Question 26

Simplify the expression: 3² × 3³

Accepted Answer

3⁵

Answer

12

Answer

6

Answer

9

Question 27

Evaluate the expression: log₂(32) + log₂(8)

Accepted Answer

6

Answer

8

Answer

2

Question 28

Which of the following is the inverse operation of exponentiation?

Accepted Answer

Logarithm

Answer

Addition

Answer

Subtraction

Answer

Multiplication

Question 29

Find the domain of the function: f(x) = log(x - 1)

Accepted Answer

(1, ∞)

Answer

(-∞, 1)

Answer

(-1, ∞)

Answer

(0, 1)

Question 30

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

Accepted Answer

Exponential curve that increases rapidly

Answer

Linear line

Answer

Exponential curve that decreases rapidly

Answer

Parabolic curve

Question 31

Which of the following applications of exponents and logarithms is important in healthcare?

Accepted Answer

Calculating drug dosages based on patient weight and response

Answer

Analyzing X-rays

Answer

Interpreting ECGs

Answer

Measuring blood pressure

Question 32

Find the value of log10(1000). Describe the significance of logarithms in healthcare.

Accepted Answer

3

Answer

0.3

Answer

0.03

Answer

30

Question 33

Which of the following is equal to 10^3? Explain the concept of powers of 10 and their applications in healthcare.

Accepted Answer

1000

Answer

0.1

Answer

100

Answer

10000

Question 34

Simplify: (x^2y)^-3. Explain the rule of exponents used in your simplification and provide an example of its application in healthcare.

Accepted Answer

x^-6y^-3

Answer

x^-3y^2

Answer

x^6y^3

Answer

x^3y^-6

Question 35

Find the value of e^0. Describe the mathematical properties of e and its significance in healthcare.

Accepted Answer

1

Answer

-1

Answer

e

Answer

0

Question 36

Which of the following is a logarithmic function? Explain the characteristics and applications of logarithmic functions in healthcare.

Accepted Answer

y = log(x)

Answer

y = e^x

Answer

y = x^2

Answer

y = sin(x)

Question 37

Solve for x: 3^(2x - 1) = 27. Explain the steps involved in your solution and discuss the importance of exponential equations in healthcare.

Accepted Answer

x = 2

Answer

x = 3

Answer

x = 1

Answer

x = 4

Question 38

Simplify the expression: 2^5 multiplied by 2^3

Accepted Answer

2^8

Answer

2^15

Answer

2^2

Answer

2^10

Question 39

Solve for x: 3^x equals 27

Accepted Answer

3

Answer

1/3

Answer

9

Question 40

Find the value of log base 10 of 1000

Accepted Answer

3

Answer

0.3

Answer

100

Answer

-3

Question 41

Which of the following is the inverse function of the exponential function y equals 2^x?

Accepted Answer

y equals log base 2 of x

Answer

y equals 1/2^x

Answer

y equals 2x

Answer

y equals x^2

Question 42

Which of the following graphs represents the function f(x) equals 2^x plus 1?

Accepted Answer

A graph that is a translated version of the exponential function y = 2^x, shifted up by one unit.

Answer

A graph that is a translated version of the exponential function y = 2^x, shifted down by one unit.

Answer

A graph that is a reflection of the exponential function y = 2^x over the x-axis.

Question 43

Which of the following applications of exponents and logarithms is relevant to Healthcare?

Accepted Answer

Calculating drug dosages based on patient weight and metabolism.

Answer

Predicting population growth.

Answer

Determining the age of fossils using carbon dating.