Logarithmic Functions: Master the Art of Solving Log Equations

Question 1

Convert the exponential equation y = 100 into its equivalent logarithmic form.

Accepted Answer

y = 10log(100)

Answer

y = log(10)100

Answer

y = log(100)10

Answer

y = 100log(10)

Question 2

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

Accepted Answer

x = 13

Answer

x = 93

Answer

x = -7

Answer

x = -93

Question 3

Which of the following represents a logarithmic function in its basic form?

Accepted Answer

y = log(x)

Answer

y = x^log

Answer

y = log(log(x))

Answer

y = x * log(x)

Question 4

Solve for x in the logarithmic equation: log₅(x + 3) = 2

Accepted Answer

x = 27

Answer

x = 12

Answer

x = 6

Answer

x = 3

Question 5

Which logarithmic property is represented by log(a) + log(b)?

Accepted Answer

log(ab)

Answer

log(a - b)

Answer

log(a/b)

Answer

log(b/a)

Question 6

If log₃(x) = y, express x in terms of y.

Accepted Answer

3^y

Answer

3^{-y}

Answer

y^3

Answer

y^{-3}

Question 7

Determine the inverse function of f(x) = log₅(x).

Accepted Answer

f⁻¹(x) = 5^x

Answer

f⁻¹(x) = log₂(5x)

Answer

f⁻¹(x) = x^5

Answer

f⁻¹(x) = 5⁻^x

Question 8

The graph of the function f(x) = log₄(x) has a vertical asymptote at x = ?

Accepted Answer

0

Answer

1

Answer

4

Question 9

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

Accepted Answer

x > -2

Answer

x > 0

Answer

x > 2

Question 10

Which of the following is the inverse function of f(x) = 5^x?

Accepted Answer

f^(-1)(x) = log₅(x)

Answer

f^(-1)(x) = x^5

Answer

f^(-1)(x) = log(x)/log(5)

Question 11

The logarithmic form of the equation y = 2^x is:

Accepted Answer

x = log2(y)

Answer

y = logx(2)

Answer

log2(x) = y

Answer

x = 2logy

Question 12

The graph of the function f(x) = log2(x) is:

Accepted Answer

Logarithmic, increasing slowly

Answer

Linear, with a slope of 1

Answer

Exponential, increasing rapidly

Answer

Parabolic, opening upward

Question 13

The equation of an exponential function with a base of 2 is:

Accepted Answer

y = 2^x

Answer

y = 2x

Answer

y = log2(x)

Answer

y = x^2

Question 14

The domain of the function f(x) = log(x-3) is:

Accepted Answer

(3, ∞)

Answer

[0, ∞)

Answer

(-∞, ∞)

Answer

(3, 0]

Question 15

The inverse function of f(x) = log2(x) is:

Accepted Answer

f^-1(x) = 2^x

Answer

f^-1(x) = 1/log2(x)

Answer

f^-1(x) = logx(2)

Answer

f^-1(x) does not exist

Question 16

Identify the graph that corresponds to the logarithmic function f(x) = log2(x + 1).

Accepted Answer

[Image of the graph of f(x) = log2(x + 1)]

Answer

[Image of the graph of f(x) = 2^x + 1]

Answer

[Image of the graph of f(x) = 1/log2(x + 1)]

Answer

[Image of the graph of f(x) = log2(x) + 1]

Question 17

Which of the following is equivalent to log_b(x) + log_b(y)?

Accepted Answer

log<sub>b</sub>(xy)

Answer

log_b(x - y)

Answer

log_b(x + y)

Answer

log<sub>b</sub>(x/y)

Question 18

What is the value of log_10(100)?

Accepted Answer

2

Answer

1

Answer

10

Answer

100

Question 19

Which of the following properties of logarithms is used to solve the equation log_3(x + 2) = log_3(4x - 5)?

Accepted Answer

log_a(b) - log_a(c) = log_a(b/c)

Answer

log_a(b) + log_a(c) = log_a(bc)

Answer

log_a(b^c) = c log_a(b)

Answer

1 + log_a(b) = log_a(b + 1)

Question 20

A certain population of bacteria doubles every hour. If there are initially 100 bacteria, how many bacteria will there be after 3 hours?

Accepted Answer

800

Answer

100

Answer

200

Answer

400

Question 21

Which of the following is the equation of the horizontal asymptote of the function f(x) = 3 ^ (x - 1) + 2?

Accepted Answer

y = 2

Answer

x = 1

Answer

x = 2

Answer

y = 3

Question 22

Which of the following is the graph of the function f(x) = log2(x - 1)?

Accepted Answer

Image of a graph that shifts the parent function f(x) = log2(x) one unit to the right.

Answer

Image of a graph that shifts the parent function f(x) = log2(x) two units up.

Answer

Image of a graph that shifts the parent function f(x) = log2(x) one unit to the left.

Question 23

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

Accepted Answer

x ∈ (-∞, -2) ∪ (-2, 2) ∪ (2, ∞)

Answer

All real numbers

Answer

x ∈ (-2, 2)

Answer

x ∈ [0, ∞)

Question 24

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

Accepted Answer

f^-1(x) = log2(x)

Answer

f^-1(x) = x^2

Answer

f^-1(x) = x / 2

Answer

f^-1(x) = 2 * x

Question 25

Simplify the expression: log2(8) - log2(2) + log2(16).

Accepted Answer

4

Answer

16

Answer

2

Answer

8

Question 26

Which of the following is an example of an application of logarithmic functions in real life?

Accepted Answer

Measuring the intensity of earthquakes using the Richter scale

Answer

Solving quadratic equations

Answer

Calculating the speed of light

Question 27

Which of the following inequalities is equivalent to log(x - 2) < 1?

Accepted Answer

x - 2 < 10

Answer

x + 2 > 10

Answer

x + 2 < 10

Answer

x - 2 > 10

Question 28

What is the graph of the logarithmic function g(x) = log(x + 1) compared to the parent function f(x) = log(x)?

Accepted Answer

Shifted one unit to the left

Answer

Shifted one unit down

Answer

Shifted one unit to the right

Answer

Shifted one unit up

Question 29

What is the logarithmic form of the equation 10^3 = 1000?

Accepted Answer

log(1000) = 3

Answer

1000^3 = 10

Answer

10^1000 = 3

Answer

log(3) = 1000

Question 30

Which of the following properties of logarithms is NOT true?

Accepted Answer

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

Answer

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

Answer

loga(a) = 1

Answer

log(1) = 0

Question 31

What is the domain of the logarithmic function h(x) = log(x - 3)?

Accepted Answer

x > 3

Answer

x < 3

Answer

x < 0

Answer

x > 0

Question 32

Which of the following equations is logarithmic?

Accepted Answer

7^x = 49

Answer

sin(x) = 0.5

Answer

x^2 - 4 = 0

Answer

x + 3 = 10

Question 33

What is the simplified form of log(100) - log(25)?

Accepted Answer

1

Answer

0.5

Answer

2

Answer

4

Question 34

Calculate the value of log₂ 16, where log represents the logarithmic function with base 2.

Accepted Answer

4

Answer

2

Answer

8

Answer

32

Question 35

Express the logarithmic expression log₃ 27 in exponential form.

Accepted Answer

3³ = 27

Answer

log₃ 27 = 3

Answer

3² = 27

Answer

27³ = 3

Question 36

Consider the logarithmic function y = log₄ x. If the graph of this function is shifted 3 units to the right and 2 units down, what is the equation of the transformed graph?

Accepted Answer

y = log₄(x - 3) - 2

Answer

y = log₄(x + 3) + 2

Answer

y = log₄(x - 3) + 2

Answer

y = log₄(x + 3) - 2

Question 37

Identify the logarithmic property that does NOT hold true:

Accepted Answer

logₐ (x + y) = logₐ x + logₐ y

Answer

logₐ (xy) = logₐ x + logₐ y

Answer

logₐ (x/y) = logₐ x - logₐ y

Answer

logₐ xⁿ = n logₐ x

Question 38

For the function f(x) = log₂ (x - 3), determine its domain.

Accepted Answer

{x | x > 3}

Answer

{x | x ≥ 3}

Answer

{x | x ≤ 3}

Answer

All real numbers

Question 39

Identify the inverse function of f(x) = 3ˣ.

Accepted Answer

g(x) = log₃ x

Answer

g(x) = x³

Answer

g(x) = x/3

Answer

g(x) = 3⁻ˣ

Question 40

Calculate the value of log₁₀ 1000.

Accepted Answer

3

Answer

10

Answer

1

Answer

100

Question 41

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

Accepted Answer

A graph with a vertical asymptote at x = 0 and a horizontal asymptote at y = 0, increasing from left to right.

Answer

A graph with a vertical asymptote at x = 0 and a horizontal asymptote at y = 1, increasing from left to right.

Answer

A graph with a vertical asymptote at x = 1 and a horizontal asymptote at y = 1, decreasing from left to right.

Answer

A graph with a vertical asymptote at x = 1 and a horizontal asymptote at y = 0, decreasing from left to right.

Question 42

Which of the following is equivalent to log(a) + log(b)?

Accepted Answer

log(ab)

Answer

log(a + b)

Answer

log(a/b)

Answer

log(a²) + log(b²)

Question 43

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

Accepted Answer

[H⁺] = 10⁻⁴

Answer

[H⁺] = 10⁻³

Answer

[H⁺] = 10⁴

Answer

[H⁺] = 10³

Question 44

Which of the following is the inverse function of f(x) = log₅(x)?

Accepted Answer

f⁻¹(x) = 5ˣ

Answer

f⁻¹(x) = 5⁻ˣ

Answer

f⁻¹(x) = log₅⁻¹(x)

Answer

f⁻¹(x) = x⁵

Question 45

What is the solution to the inequality log₃(x + 2) < 1?

Accepted Answer

0 < x < 2

Answer

x < 0

Answer

x < 2

Answer

x > 2

Question 46

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

Accepted Answer

x = 3

Answer

x = 5

Answer

x = 25

Answer

x = 125

Question 47

What is the graph of the logarithmic function f(x) = log₂(x + 3) like?

Accepted Answer

Increasing, passes through the point (-2, 0)

Answer

Decreasing, passes through the point (-3, 0)

Answer

Increasing, passes through the point (0, 0)

Answer

Decreasing, passes through the point (0, 1)

Question 48

Which of the following properties of logarithms is used to simplify log(x / y) to log(x) - log(y)?

Accepted Answer

Quotient Property

Answer

Power Property

Answer

Product Property

Answer

Change of Base Property

Question 49

Find the domain of the logarithmic function f(x) = ln(x² - 1).

Accepted Answer

x > 1 or x < -1

Answer

x > -1

Answer

x < 1

Answer

x > 0

Question 50

What is the inverse function of f(x) = log2(x)?

Accepted Answer

f^-1(x) = 2^x

Answer

f^-1(x) = e^x

Answer

f^-1(x) = x^2

Answer

f^-1(x) = log10(x)

Question 51

Which of the following is an example of a real-world application of logarithmic functions?

Accepted Answer

Measuring the intensity of earthquakes

Answer

Estimating population growth

Answer

Calculating the trajectory of a projectile

Answer

Predicting weather patterns

Question 52

Find the value of log5(125) + log5(5).

Accepted Answer

4

Answer

2

Answer

6

Answer

8

Question 53

Which of the following logarithmic equations has a solution of x = 2?

Accepted Answer

log4(x) = 1/2

Answer

log2(x) = 4

Answer

log(x - 2) = 1

Answer

5logx = 2

Question 54

What is the range of the logarithmic function f(x) = -log(x + 1)?

Accepted Answer

(-∞, 0]

Answer

[0, 1]

Answer

[0, ∞)

Answer

(-∞, +∞)