MCQ Test- Concept and Application of Logarithm (Class- 9, Experience- 3)

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Multiple Choice Quiz: Test Your Knowledge!

Question 1: If log_b(xy)=log_b(x)+log_b(y), what property of logarithms does this represent?

A. Product Rule
B. Quotient Rule
C. Power Rule
D. Change of Base Formula

Question 2: What is the base of the common logarithm (log)?

A. 2
B. e
C. 10
D. 1

Question 3: What is the base of the natural logarithm (ln)?

A. 2
B. e
C. 1
D. 10

Question 4: Simplify log(100).

A. 0
B. 1
C. 2
D. 3

Question 5: What is log_b(1) for any base b?

A. 0
B. 1
C. −1
D. b

Question 6: If b^y=x, what is log_b(x)?

A. y
B. x
C. b
D. 1

Question 7: Simplify log(2)+log(5).

A. log(7)
B. log(10)
C. log(25)
D. log(1)

Question 8: If log_b(16)=4, what is b?

A. 2
B. 4
C. 8
D. 16

Question 9: Solve for x: log₂(x)=5.

A. 10
B. 16
C. 25
D. 32

Question 10: If a=log_b(c), which of the following is equivalent?

A. b^a=c
B. a^b=c
C. c^a=b
D. b^c=a

Question 11: Evaluate log⁡₄(16)

A. 1
B. 2
C. 3
D. 4

Question 12: What is the logarithm of 100 to the base 10?

A. 1
B. 2
C. 3
D. 4

Question 13: If 2³=8, what is log₂8?

A. 1
B. 2
C. 3
D. 4

Question 14: What is the base of the natural logarithm?

A. 2
B. e
C. 10
D. 1

Question 15: What is the base of the common logarithm?

A. 2
B. e
C. 10
D. 1

Question 16: Which of the following represents the product rule of logarithms?

A. log_b(𝑥𝑦)= log_b(𝑥)+log_b(𝑦)
B. log_b(𝑥/𝑦)= log_b(𝑥)−log_b(𝑦)
C. log_b(𝑥𝑦)= 𝑦log_b(𝑥)
D. log_b(x)= log_c(x)log_c(b)

Question 17: What is the value of log₁₀(1)?

A. 0
B. 1
C. 10
D. Undefined

Question 18: If log_b(xy)= log_b(x)−log_b(y), what property is this?

A. Product Rule
B. Quotient Rule
C. Power Rule
D. Change of Base Formula

Question 19: Which of the following represents the power rule of logarithms?

A. log_b(xy)= log_b(x)+log_b(y)
B. log_b(x/y)= log_b(x)−log_b(y)
C. log_b(xy)= ylog_b(x)
D. log_b(x)= log_c(x)log_c(b)

Question 20: If bⁿ=a, what is log_b(a)?

A. n
B. a
C. b
D. 1

Question 21: Which of the following is the change of base formula?

A. log_b(xy)= log_b(x)+log_b(y)
B. log_b(x/y)= log_b(x)−log_b(y)
C. log_b(x)= ylog_b(x)
D. log_b(x)= log_c(x)log_c(b)

Question 22: Solve for x: 2^x=16.

A. 2
B. 3
C. 4
D. 5

Question 23: If 3^x=81, what is x?

A. 3
B. 4
C. 5
D. 6

Question 24: If 5^x=25, what is x?

A. 2
B. 3
C. 4
D. 5

Question 25: Solve for x: e^x=1.

A. 0
B. 1
C. e
D. 2

Question 26: What is log₂(32)?

A. 3
B. 4
C. 5
D. 6

Question 27: If ln(x)=2, then x=?

A. e
B. e²
C. 2
D. 1

Question 28: What is the base of the logarithm used to measure soil fertility?

A. 2
B. e
C. 10
D. None of the above

Question 29: The depreciation of an object can be calculated using which logarithmic concept?

A. Product Rule
B. Quotient Rule
C. Exponential Decay
D. Power Rule

Question 30: Find the value of x if ln(x)=0.

A. 0
B. 11
C. e
D. 1010

Question 31: What is the value of log_e(e)?

A. 0
B. 1
C. e
D. Undefined

Question 32: Simplify log(1) for any base.

A. 0
B. 1
C. Base dependent
D. Undefined

Question 33: If blog_b(x)=y, what is x?

A. y
B. e^y
C. b^y
D. b

Question 34: If a^b=c, what is log_a(c)?

A. b
B. a
C. c
D. 1

Question 35: What is the value of log₂(8)?

A. 2
B. 3
C. 4
D. 5

Question 36: Convert log₅(125) to a simpler form.

A. 3
B. 4
C. 5
D. 6

Question 37: Evaluate log₃(81).

A. 2
B. 3
C. 4
D. 5

Question 38: Solve: log¹⁰=1000.

A. 2
B. 3
C. 4
D. 5

Question 39: If log(1000)=3, what is the base?

A. 2
B. 3
C. 10
D. None of the above

Question 40: Find the value of x in the equation 3^x=27.

A. 2
B. 3
C. 4
D. 5

Question 41: The half-life of a certain substance is 5 years. If we start with 100 grams, how much will remain after 10 years?

A. 50 grams
B. 25 grams
C. 20 grams
D. 10 grams

Question 42: Which type of logarithm is used to measure the magnitude of earthquakes?

A. Common logarithm
B. Natural logarithm
C. Binary logarithm
D. None of the above

Question 43: If log₂(x)=5, what is x?

A. 10
B. 16
C. 25
D. 32

Question 44: Solve for x: log₃(x)=4.

A. 27
B. 64
C. 81
D. 243

Question 45: If e^x=20, what is x?

A. ln(20)
B. eln(20)
C. log(20)
D. None of the above

Question 46: Solve for x: log₅(x)=3.

A. 5
B. 15
C. 25
D. 125

Question 47: If log(x)=2, what is x?

A. 10
B. 20
C. 100
D. 1000

Question 48: Find x if log₃(x)=5.

A. 27
B. 81
C. 243
D. 729

Question 49: If e^x=7, what is x?

A. log(7)
B. elog(7)
C. ln(7)
D. None of the above

Question 50: Solve for x: 10^x=100.

A. 1
B. 2
C. 3
D. 4

Question 51: What is log₁₀(10000)?

A. 2
B. 3
C. 4
D. 5

Question 52: Evaluate log₄(64).

A. 2
B. 3
C. 4
D. 5

Question 53: Find the value of x if log₉(x)=2.

A. 27
B. 36
C. 81
D. 100

Question 54: Simplify log₆(216).

A. 2
B. 3
C. 4
D. 5

Question 55: Solve for x: log₅(x)=4.

A. 125
B. 500
C. 625
D. 1024

Question 56: The population of a bacteria colony doubles every 3 hours. If the initial population is 200, what will the population be after 9 hours?

A. 400
B. 600
C. 800
D. 1600

Question 57: If e^0.693≈2, what is the approximate value of ln(2)?

A. 0.5
B. 0.693
C. 1
D. 2

Question 58: The Richter scale for measuring earthquake magnitudes is logarithmic. If one earthquake has a magnitude of 6 and another has a magnitude of 8, how many times more powerful is the second earthquake?

A. 2 times
B. 10 times
C. 100 times
D. 1000 times

Question 59: The derivative of ln(x) is:

A. 1x
B. xln(x)
C. ln(x)  x
D. e^x

Question 60: The integral of 1x with respect to x is:

A. ln∣x∣+C
B. e^x+C
C. 1x + C
D. xln(x)−x+C

Question 61: If y=e^x, then dydx=?

A. e^x
B. ln(e^x)
C. xe^x-1
D. e^x-1

Question 62: The limit of ln(x)   x as x approaches infinity is:

A. 0
B. 1
C. Infinity
D. Undefined

Question 63: If f(x)=ln(x₂), then f′(x)=?

A. 2x
B. 1x
C. 2ln(x)
D. xln(x)

Question 64: The decibel scale for sound intensity uses which base for its logarithms?

A. 2
B. e
C. 10
D. 100

Question 65: What is the common logarithm of 1,000,000?

A. 5
B. 6
C. 7
D. 8

Question 66: The pH scale is used to measure acidity and is based on the logarithm of the concentration of which ion?

A. Hydroxide ion
B. Hydrogen ion
C. Sodium ion
D. Chloride ion

Question 67: Which type of logarithm is used in compound interest calculations?

A. Common logarithm
B. Natural logarithm
C. Binary logarithm
D. None of the above

Question 68: If log₂(8)=3, what is log₈(2)?

A. 13
B. 2
C. 3
D. 4

Question 69: Which of the following is an identity for logarithms?

A. log_b(1)=1
B. log_b(b)=0
C. log_b(0)=∞
D. log_b(1)=0

Question 70: Simplify log₁₀(0.01).

A. -1
B. -2
C. 1
D. 2

Question 71: Find x if 10^x=0.001.

A. -3
B. -2
C. 2
D. 3

Question 72: What is log_2³(8)?

A. 1
B. 2
C. 3
D. 4

Question 73: Simplify log_a^b(a^c).

A. bc
B. c
C. bc
D. cd

Question 74: Solve for x: ln(x)=3.

A. e³
B. 3
C. ln(3)
D. 3e

Question 75: If log_b(x)=4 and b=2, what is x?

A. 8
B. 16
C. 32
D. 64

Question 76: What is the value of ln(1e)?

A. 1
B. -1
C. 0
D. e

Question 77: Solve for x: e^2x=5.

A. ln(5)  2
B. 2ln(5)
C. ln(5)
D. ln(2)  5

Question 78: Find x if 10^x-2=1000.

A. 4
B. 5
C. 6
D. 7

Question 79: What is the derivative of y=log_a(x)?

A.   1xln(a)
B. ln(a)  x
C.  1ax
D. ax

Question 80: The equation log_b(x)=log_b(y) implies:

A. x=y
B. x≠y
C. b=x
D. b=y

Question 81: If log(x+2)=log(x−2), then:

A. x=4
B. x=2
C. x=0
D. No solution

Question 82: Simplify log(√ 10 ​).

A. log(10)
B. 12 log(10)
C. √ log(10)​ 
D. 2log(10)

Question 83: Solve for x: e^x2=10.

A. √ ln(10)​ 
B. ln(10)
C. e¹⁰
D. e⁵

Question 84: Which of the following represents the quotient rule for logarithms?

A. log_b(xy)=log_b(x)+log_b​(y)
B. log_b(xy)=log_b(x)−log_b(y)
C. log_b​(x^y)=ylog_b(x)
D. log_b(x)=log_c​(x)​log_c(b)

Question 85: What is log₂(16)?

A. 2
B. 3
C. 4
D. 5

Question 86: Find the value of log₃(81).

A. 2
B. 3
C. 4
D. 5

Question 87: Evaluate log₅(125).

A. 2
B. 3
C. 4
D. 5

Question 88: Solve for x: log₄(64)=x.

A. 2
B. 3
C. 4
D. 5

Question 89: If log₇(x)=2, then x=?

A. 14
B. 28
C. 49
D. 64

Question 90: The natural logarithm of 1 is:

A. 0
B. 1
C. e
D. ∞

Question 91: Which of the following represents the power rule for logarithms?

A. log_b(xy)=log_b(x)+log_b(y)
B. log_b(xy)=log_b(x)−log_b(y)
C. log_b(x^y)=ylog_b(x)
D. log_b(x)=log_c(x)log_c(b)

Question 92: The base of the natural logarithm ln(x) is:

A. 10
B. e
C. 2
D. x

Question 93: The value of e^ln(5) is:

A. 0
B. 1
C. 5
D. e

Question 94: In the Richter scale, the magnitude M of an earthquake is given by M=log₁₀( AA₀), where A is the amplitude of the seismic waves and A₀ is a reference amplitude. If the amplitude of a seismic wave is 10⁵ times the reference amplitude, what is the magnitude of the earthquake?

A. 4
B. 5
C. 6
D. 7

Question 95: What is the change of base formula for logarithms?

A. log_b(xy)=log_b(x)+log_b(y)
B. log_b(xy)=log_b(x)−log_b(y)
C. log_b(x^y)=ylog_b(x)
D. log_b(x)=log_c(x)log_c(b)

Question 96: The population of a certain city doubles every 10 years. If the current population is 50,000, what will the population be in 30 years?

A. 100000
B. 200000
C. 400000
D. 800000

Question 97: In exponential growth, the population P(t) at time t is given by P(t)=P₀e^kt. If P₀=100 and k=0.02, what is P(50)?

A. 100
B. 182
C. 270
D. 400

Question 98: The half-life of a radioactive isotope is 20 years. How much of a 100 gram sample will remain after 60 years?

A. 12.5 grams
B. 25 grams
C. 50 grams
D. 75 grams