Logarithms: Master the Laws and Solve Equations with Confidence

Question 1

Identify the logarithmic function:

Accepted Answer

y = log(x)

Answer

y = 2^x

Answer

y = x^2

Answer

y = sin(x)

Question 2

Evaluate **log10(100)**.

Accepted Answer

2

Answer

100

Answer

1

Question 3

Simplify the following expression: **log(xy^2)**.

Accepted Answer

logx + 2logy

Answer

2log(xy)

Answer

log(x^2y)

Answer

logy^2

Question 4

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

Accepted Answer

[2, ∞)

Answer

[-2, ∞)

Answer

(-∞, 2]

Answer

(2, ∞)

Question 5

Describe the key features of the graph of **y = log(x)**.

Accepted Answer

It has a vertical asymptote at x = 0 and is undefined for x ≤ 0.

Answer

It has a horizontal asymptote at y = 0 and is undefined for x ≥ 0.

Answer

It is symmetric about the y-axis.

Question 6

What is the inverse function of **y = log2(x)**.

Accepted Answer

x = 2^y

Answer

x = y^2

Answer

x = log(2y)

Answer

x = 2/y

Question 7

Which expression is equivalent to **log(a) + log(b)**?

Accepted Answer

log(ab)

Answer

log(a^b)

Answer

log(b/a)

Answer

log(a/b)

Question 8

Which of the following is the definition of a logarithm?

Accepted Answer

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

Answer

The difference between two numbers

Answer

The number that is being raised to a power

Question 9

Solve for x: log3(x) = 2

Accepted Answer

9

Answer

2

Answer

4

Question 10

Which of the following represents the equation y = log(x - 2) in exponential form?

Accepted Answer

x - 2 = 10^y

Answer

log(y) = x - 2

Answer

x - 2 = y^10

Answer

10^x - 2 = y

Question 11

Which of the following is an example of the application of logarithms?

Accepted Answer

Measuring the acidity of a solution (pH)

Answer

Solving quadratic equations

Answer

Calculating the area of a triangle

Answer

Finding the velocity of a moving object

Question 12

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

Accepted Answer

x > 3

Answer

All real numbers

Answer

x < 3

Answer

x > 0

Question 13

What is the range of the function y = log(x - 1) + 4?

Accepted Answer

y > 4

Answer

y < 4

Answer

0 < y < 10

Question 14

Which of the following graphs represents the function y = log(x + 1)?

Accepted Answer

Image showing a graph of y=log(x+1), an increasing curve with a vertical asymptote at x=-1

Answer

Image showing a graph of y=logx, an increasing curve with a vertical asymptote at x=0

Answer

Image showing a graph of y=x^2, a parabola opening upwards

Question 15

Which of the following best defines a logarithm?

Accepted Answer

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

Answer

The number of times a number must be divided by itself to obtain a given value

Answer

The process of multiplying a number by itself to obtain a given value

Question 16

What is the value of log₂ 16?

Accepted Answer

4

Answer

2

Answer

16

Answer

8

Question 17

Which of the following is the inverse operation of taking the logarithm of a number?

Accepted Answer

Exponentiation

Answer

Division

Answer

Square root

Answer

Multiplication

Question 18

Solve for x: log(x²) - log(x) = 1

Accepted Answer

x = 10

Answer

x = 1 or x = 10

Answer

x = 5

Answer

x = 2

Question 19

Which of the following equations represents exponential growth, where the variable 'y' represents the final amount, 'x' is the initial amount, 'r' is the growth rate, and 't' is the time?

Accepted Answer

y = x * (1 + r)^t

Answer

y = x + r*t

Answer

y = x * r^t

Answer

y = x * (1 - r)^t

Question 20

Simplify the expression: log₅ 125 + log₅ 5

Accepted Answer

log₅ 625

Answer

log₅ 25

Answer

log₅ 100

Answer

5 log₅ 5

Question 21

Which of the following properties of logarithms is NOT valid?

Accepted Answer

logₐ (b - c) = logₐ b - logₐ c

Answer

logₐ bⁿ = n logₐ b

Answer

logₐ (b/c) = logₐ b - logₐ c

Answer

logₐ b^c = c logₐ b

Question 22

What is the value of log₁₀ 0.001?

Accepted Answer

-3

Answer

1

Answer

-1

Answer

3