Logarithms: A Comprehensive Guide for Sixth Form Students

Question 1

Identify the inverse function of f(x) = logₐ(x).

Accepted Answer

g(x) = aˣ

Answer

g(x) = xᵃ

Answer

g(x) = logᵃ(x)

Answer

g(x) = √x

Question 2

Use logarithms to solve the exponential equation: 2^x = 5

Accepted Answer

x = log₂(5)

Answer

x = log₅(2)

Answer

x = 5^2

Answer

x = 2^5

Question 3

Define the logarithm of a number.

Accepted Answer

The exponent to which the base must be raised to obtain that number.

Answer

The number that must be multiplied by the base to obtain that number.

Answer

The reciprocal of that number.

Answer

The square root of that number.

Question 4

The pH of a solution is a measure of its acidity and is calculated as the negative logarithm of the hydrogen ion concentration. If the pH is 3, what is the hydrogen ion concentration?

Accepted Answer

10^-3

Answer

10^3

Answer

log(-3)

Answer

-log(3)

Question 5

What is the inverse function of the logarithmic function?

Accepted Answer

Exponential function

Answer

Trigonometric function

Answer

Inverse trigonometric function

Answer

Polynomial function

Question 6

The derivative of y = log(x) is:

Accepted Answer

1/x

Answer

-1/x

Answer

x

Answer

-x

Question 7

Use a logarithmic law to expand log⁡ 2(x^2 + 3x).

Accepted Answer

Product rule: log⁡ a(bc) = log⁡ a(b) + log⁡ a(c)

Answer

Quotient rule: log⁡ a(b/c) = log⁡ a(b) - log⁡ a(c)

Answer

Power rule: log⁡ a(b^c) = c * log⁡ a(b)

Answer

Change of base rule: log⁡ a(b) = log⁡ c(b) / log⁡ c(a)

Question 8

Which of the following can be the base of a logarithmic function?

Accepted Answer

any positive real number except 1

Answer

0

Answer

1

Answer

any negative real number

Question 9

The pH of a solution is measured on a logarithmic scale, with a lower pH indicating a more acidic solution. If the pH is 4, what is the hydrogen ion concentration in M?

Accepted Answer

10^-4 M

Answer

10^-6 M

Answer

10^-8 M

Answer

10^-10 M

Question 10

Which of the following accurately defines the logarithm of a number b to the base a?

Accepted Answer

The exponent to which 'a' must be raised to obtain 'b'

Answer

The base that, when exponentiated to the power 'b', yields 'a'

Answer

The number that, when multiplied by 'b', gives 'a'

Answer

The inverse function of exponentiation

Question 11

Which function is the inverse of the logarithmic function y = log_a(x)?

Accepted Answer

y = ax

Answer

y = xa

Answer

y = ex

Answer

y = 10x

Question 12

Which of the following is NOT an application of logarithmic functions?

Accepted Answer

Solving systems of simultaneous equations

Answer

Solving exponential equations

Answer

Measuring earthquake intensity

Answer

Calculating pH of solutions

Question 13

Calculate the value of log₃(27).

Accepted Answer

6

Answer

3

Answer

9

Answer

18

Question 14

For the logarithmic function y = log_a(x - 2), what is the appropriate domain?

Accepted Answer

x ≠ 2

Answer

x < 2

Answer

x > 2

Answer

x > 0

Question 15

Convert the exponential equation 10^x = 1000 to its logarithmic form.

Accepted Answer

log₁₀(1000) = x

Answer

x = 3

Answer

x = 1000

Answer

log₁₀(x) = 1000

Question 16

Which of the following is NOT an application of logarithms?

Accepted Answer

Solving quadratic equations

Answer

Measuring sound intensity

Answer

Finding the pH of a solution

Answer

Calculating half-lives of radioactive substances

Question 17

What is the domain of the function f(x) = log₁₀(x - 1)?

Accepted Answer

x > 1

Answer

x < 1

Answer

x = 1

Answer

All real numbers

Question 18

Identify the logarithmic inequality:

Accepted Answer

log₂(x) > 3

Answer

2x < 5

Answer

x > log₂(3)

Answer

log₂(x) + log₂(y) < log₂(z)

Question 19

Which of the following is the logarithm of x to the base b written in mathematical notation?

Accepted Answer

logbx

Answer

xb

Answer

bx

Answer

xlogb

Question 20

What is the value of log₂(32)?

Accepted Answer

4

Answer

2

Answer

8

Answer

16

Question 21

Which logarithmic property allows us to rewrite log(ab) as log(a) + log(b)?

Accepted Answer

Product rule

Answer

Quotient rule

Answer

Power rule

Answer

Change of base rule

Question 22

Solve for x: log_x(27) = 3

Accepted Answer

9

Answer

3

Answer

27

Answer

81

Question 23

What is the inverse function of y = log_b(x)?

Accepted Answer

y = bx

Answer

y = ex

Answer

y = 10x

Answer

y = ln(x)

Question 24

Calculate the logarithm of 100 to the base 5.

Accepted Answer

2

Answer

5

Answer

10

Answer

20

Question 25

Simplify the expression: log(x^2 * y^3)

Accepted Answer

2logx + 3logy

Answer

logx^2 + logy^3

Answer

x^2logy + y^3logx

Answer

log(x + y)

Question 26

Which equation is equivalent to 2^x = 8?

Accepted Answer

x = log2(8)

Answer

x = 8log2

Answer

x = 2^8

Answer

x = 3

Question 27

Solve the equation: loge(x + 1) = 2

Accepted Answer

x = e^2 - 1

Answer

x = 2e - 1

Answer

x = e^2 + 1

Answer

x = 2 - e

Question 28

Find the inverse of the function y = log5(x)

Accepted Answer

y = 5^x

Answer

x = 5^y

Answer

x = log5(y)

Answer

y = log5(x)

Question 29

Which of the following statements is NOT a property of logarithms?

Accepted Answer

Logarithms are defined only for positive numbers.

Answer

Logarithms are the inverse functions of exponentials.

Answer

Logarithms can be used to solve exponential equations.

Answer

The base of a logarithm can be any positive number.

Question 30

The pH of a solution is defined as -log₁₀[H+], where [H+] is the molar concentration of hydrogen ions in the solution. What is the pH of a solution with [H+] = 10^-4?

Accepted Answer

4

Answer

10

Answer

14

Answer

20

Question 31

Which of the following is the graph of the function y = log₂(x)?

Accepted Answer

A straight line with positive gradient

Answer

A parabola opening upwards

Answer

A parabola opening downwards

Answer

A hyperbola.

Question 32

Which of the following is the domain of the function f(x) = log0.5(x - 3)?

Accepted Answer

x > 3

Answer

x < 3

Answer

x ≥ 3

Answer

x ≤ 3

Question 33

Which of the following is the range of the function g(x) = -3 + logx(4)?

Accepted Answer

-3 ≤ x ≤ ∞

Answer

All real numbers

Answer

x > 0

Answer

x < 0

Question 34

What is the definition of a logarithm?

Accepted Answer

The exponent to which a base must be raised to produce a given number

Answer

The inverse operation of multiplication

Answer

The square root of a number

Answer

The exponential of a number

Question 35

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

Accepted Answer

(3, ∞)

Answer

(0, ∞)

Answer

(-∞, 0)

Answer

(-∞, 3)

Question 36

What is the inverse function of y = log_a(x)?

Accepted Answer

y = ax

Answer

y = xa

Answer

y = log_x(a)

Answer

y = a-x

Question 37

Which of the following is the logarithmic form of the equation 5^x = 125?

Accepted Answer

log₅(125) = x

Answer

5log(x) = 125

Answer

log(125/5) = x

Answer

125^x = 5

Question 38

Simplify the expression: log(4x^3 * y^5)

Accepted Answer

3log(x) + 5log(y)

Answer

4x^3log(y^5)

Answer

log(3x + 5y)

Answer

log(4x^3y^5)

Question 39

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

Accepted Answer

f⁻¹(x) = log₃(x)

Answer

f⁻¹(x) = 1/3^x

Answer

f⁻¹(x) = x/3

Answer

f⁻¹(x) = 3x

Question 40

A cell phone signal decays exponentially over time. The signal strength S, in decibels (dB), after t minutes is given by the formula: S(t) = -10log(t) + 60. What is the signal strength after 10 minutes?

Accepted Answer

40 dB

Answer

50 dB

Answer

30 dB

Answer

20 dB

Question 41

What is the value of log₂ 8?

Accepted Answer

3

Answer

4

Answer

2

Answer

6

Question 42

If log₃ x = 2, what is the value of x?

Accepted Answer

9

Answer

1/3

Answer

6

Answer

1/9

Question 43

Solve the equation: log₂ (x + 1) = 3

Accepted Answer

7

Answer

9

Answer

1

Answer

5

Question 44

The intensity of sound is measured in decibels (dB). The formula for decibel level is L = 10 log₁₀ (I/I₀), where I is the intensity of the sound and I₀ is the reference intensity. If the intensity of a sound doubles, what is the increase in decibel level?

Accepted Answer

Approximately 3 dB

Answer

Approximately 20 dB

Answer

Approximately 10 dB

Answer

Approximately 1 dB

Question 45

What is the value of log₁₀ 0.01?

Accepted Answer

-2

Answer

2

Answer

1/2

Answer

-1

Question 46

Simplify the expression: log₅ (25x²)

Accepted Answer

2 + 2log₅ x

Answer

2log₅ 5 + log₅ x

Answer

5 + 2log₅ x

Answer

2log₅ (25x)

Question 47

The population of a town is growing exponentially. The formula for population growth is P = P₀e^(kt), where P is the population at time t, P₀ is the initial population, k is the growth constant, and e is the base of natural logarithms. If the population doubles every 10 years, what is the value of the growth constant k?

Accepted Answer

k = ln(2)/10

Answer

k = ln(10)/2

Answer

k = 2/10

Answer

k = 10ln(2)

Question 48

Solve the equation: 3^(x+2) = 27

Accepted Answer

x = 1

Answer

x = 3

Answer

x = -1

Answer

x = 0

Question 49

What is the value of log₂ 16?

Accepted Answer

4

Answer

2

Answer

1/2

Answer

8

Question 50

A population of bacteria doubles every hour. If the initial population is 1000, what will the population be after 3 hours? (Use the formula: P = P₀ * 2^(t/h), where P is the final population, P₀ is the initial population, t is the time, and h is the doubling time.)

Accepted Answer

8000

Answer

4000

Answer

2000

Question 51

Which of the following is the graph of y = log₂ x?

Accepted Answer

A graph with an increasing curve that passes through (1,0) and approaches the y-axis asymptotically.

Answer

A graph with a decreasing curve that passes through (1,0) and approaches the x-axis asymptotically.

Answer

A straight line with a positive slope passing through the origin.

Question 52

What is the domain of the function f(x) = log₃ (x - 2)?

Accepted Answer

x > 2

Answer

x < 2

Answer

x ≥ 2

Answer

all real numbers