Logarithms: Properties, Equations, and Applications

Question 1

What is the inverse exponential function of the logarithmic function f(x) = log5(x)?

Accepted Answer

f^-1(x) = 5^x

Answer

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

Answer

f^-1(x) = x^5

Answer

f^-1(x) = 5/x

Question 2

What is the definition of the _logarithm_ of a number ( x ), with base ( a )?

Accepted Answer

The _exponent_ to which ( a ) must be raised to obtain ( x ).

Answer

The number whose _antilogarithm_ is ( x ).

Answer

The _product_ of ( a ) and ( x ).

Answer

The _quotient_ of ( x ) and ( a ).

Question 3

What is the _inverse function_ of the logarithm?

Accepted Answer

Exponential function

Answer

Power function

Answer

Trigonometric function

Answer

Logarithmic function

Question 4

What is the value of ( log_{10} 100 )?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 5

Simplify the expression: ( log_a sqrt{x} ).

Accepted Answer

( rac{1}{2} log_a x ).

Answer

( sqrt{log_a x} ).

Answer

( 2 log_a x ).

Answer

( (None of the above.) )

Question 6

Solve the logarithmic equation: log(x + 2) = 3

Accepted Answer

x = 7

Answer

x = 1

Answer

x = 9

Answer

x = 11

Question 7

Simplify the logarithmic expression: log(100) + log(10)

Accepted Answer

2log(10)

Answer

log(110)

Answer

log(1010)

Answer

2 + log(10)

Question 8

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

Accepted Answer

A curve that increases from left to right

Answer

A curve that decreases from left to right

Answer

A straight line with positive slope

Answer

A straight line with negative slope

Question 9

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

Accepted Answer

y = 10^x

Answer

y = e^x

Answer

y = x^10

Answer

y = x^e

Question 10

Simplify the logarithmic expression: log(a^m * b^n)

Accepted Answer

mlog(a) + nlog(b)

Answer

log(a) / log(b)

Answer

m + n

Answer

m * n

Question 11

Convert the exponential equation 10^x = y into logarithmic form.

Accepted Answer

log(y) = x

Answer

log(x) = y

Answer

log(10) = xy

Answer

log(y) = 10x

Question 12

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

Accepted Answer

x > 2

Answer

x < 2

Answer

x ≥ 2

Answer

x ≤ 2

Question 13

Which of the following graphs represents the function: y = logₒ(x)?

Accepted Answer

Exponential curve

Answer

Upward parabola

Answer

Downward parabola

Answer

Linear curve

Question 14

Find the domain of the function f(x) = log(x - 3).

Accepted Answer

x > 3

Answer

x > 0

Answer

x < 3

Answer

x < 0

Question 15

If log(x) = a and log(y) = b, express log(x^2 * y^3) in terms of a and b.

Accepted Answer

2a + 3b

Answer

2a - 3b

Answer

a + b

Question 16

What is the logarithm of 1000 to the base 10?

Accepted Answer

3

Answer

10

Answer

0.3

Answer

0.1

Question 17

Solve the equation: log₂ (x + 3) = 5

Accepted Answer

31

Answer

24

Answer

32

Answer

28

Question 18

Find the value of log₆⁴ 4

Accepted Answer

3

Answer

2

Answer

4

Answer

1/3

Question 19

Simplify the expression: log₄ (16x²) - 2 log₄ (2x)

Accepted Answer

3

Answer

1

Answer

5

Answer

2

Question 20

What is the value of ln e?

Accepted Answer

1

Answer

0

Answer

e

Answer

-1

Question 21

Solve the equation: e^(2x) = 10

Accepted Answer

ln(10)/2

Answer

2/10

Answer

10/2

Answer

log₁₀(10)/2

Question 22

Find the domain of the function: f(x) = log₅ (x - 2)

Accepted Answer

x > 2

Answer

x < 2

Answer

x ≤ 2

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

log₁₀(100) = 2

Answer

ln(e²) = 2

Question 24

Use logarithms to solve the equation: 2³ = 16

Accepted Answer

4

Answer

3

Answer

6

Answer

5

Question 25

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

Accepted Answer

(Image of a logarithmic graph)

Answer

(Image of a linear graph)

Answer

(Image of an exponential graph)

Answer

(Image of a trigonometric graph)