Unit 2 Logarithm
Mcqs
Here are some multiple-choice questions (MCQs) on logarithms:
_Basic Logarithms_
1. What is the value of log₂(8)?
A) 2
B) 3
C) 4
D) 5
Answer: B) 3
1. Which of the following is equivalent to log(100)?
A) log(10²)
B) log(10³)
C) log(10⁴)
D) log(10⁵)
Answer: A) log(10²)
1. What is the base of the common logarithm?
A) 2
B) 10
C) e
D) π
Answer: B) 10
_Logarithmic Properties_
1. What is log(ab)?
A) log(a) + log(b)
B) log(a) - log(b)
C) log(a) × log(b)
D) log(a) / log(b)
Answer: A) log(a) + log(b)
1. What is log(a/b)?
A) log(a) + log(b)
B) log(a) - log(b)
C) log(a) × log(b)
D) log(a) / log(b)
Answer: B) log(a) - log(b)
1. What is log(a²)?
A) 2log(a)
B) log(a)²
C) log(a) / 2
D) 2 / log(a)
Answer: A) 2log(a)
_Exponential and Logarithmic Equations_
1. What is the solution to 2^x = 16?
A) x = 2
B) x = 3
C) x = 4
D) x = 5
Answer: C) x = 4
1. What is the solution to log(x) = 2?
A) x = 10²
B) x = 10³
C) x = e²
D) x = π²
Answer: A) x = 10²
1. What is the solution to e^x = 10?
A) x = ln(10)
B) x = log(10)
C) x = 10
D) x = e
Answer: A) x = ln(10)
_Logarithmic Inequalities_
1. What is the solution to log(x) > 2?
A) x > 10²
B) x < 10²
C) x = 10²
D) x ≠ 10²
Answer: A) x > 10²
1. What is the solution to log(2x) < 3?
A) x < 10³ / 2
B) x > 10³ / 2
C) x = 10³ / 2
D) x ≠ 10³ / 2
Answer: A) x < 10³ / 2
Challenge questions
Here are some challenge questions on logarithms:
*Basic Logarithms*
1. Evaluate: log₂(8)
2. Simplify: log(100) - log(10)
3. Find x: log₅(x) = 2
*Logarithmic Properties*
1. Prove: log(a) + log(b) = log(ab)
2. Simplify: log(2x) + log(3x) using logarithmic properties
3. Evaluate: log(4) / log(2)
*Exponential and Logarithmic Equations*
1. Solve: 2^x = 16
2. Solve: log(x) + log(2x) = 2
3. Find x: e^(2x) = 10
*Logarithmic Inequalities*
1. Solve: log(x) > 2
2. Solve: log(2x) < 3
3. Find x: log(x) ≤ 0
*Applications of Logarithms*
1. Find the pH of a solution with a hydrogen ion concentration of 0.0001 M.
2. Calculate the magnitude of an earthquake with a seismic moment of 10^22
Nm.
3. Determine the number of decibels (dB) for a sound with an intensity of 10^-4
W/m².
*Challenge Problems*
1. Prove: log(a/b) = log(a) - log(b)
2. Solve: x^log(x) = 10
3. Find the value of x: log(2x) = log(x) + log(2)
Antilog
Here are some challenge questions with solutions on antilog:
*Basic Antilog Questions*
1. Find antilog(2.3)
Solution:
antilog(2.3) = 10^2.3 ≈ 199.5262
1. Evaluate antilog(0.5)
Solution:
antilog(0.5) = 10^0.5 ≈ 3.1622777
1. Simplify antilog(log(x))
Solution:
antilog(log(x)) = x
*Antilog Properties*
1. Prove: antilog(a) × antilog(b) = antilog(a + b)
Solution:
antilog(a) × antilog(b) = 10^a × 10^b = 10^(a + b) = antilog(a + b)
1. Simplify: antilog(2log(x))
Solution:
antilog(2log(x)) = antilog(log(x^2)) = x^2
1. Evaluate: antilog(log(2x))
Solution:
antilog(log(2x)) = 2x
*Antilog Applications*
1. Find the value of x: antilog(x) = 50
Solution:
10^x = 50
x = log(50) ≈ 1.69897
1. Evaluate: antilog(3log(2))
Solution:
antilog(3log(2)) = 2^3 = 8
1. Simplify: antilog(log(a/b))
Solution:
antilog(log(a/b)) = a/b
*Challenge Problems*
1. Solve: antilog(x) + antilog(2x) = 100
Solution:
10^x + 10^(2x) = 100
Let 10^x = y
y + y^2 = 100
y^2 + y - 100 = 0
(y + 11)(y - 10) = 0
y = 10 (reject y = -11)
10^x = 10
x=1
1. Find the value of x: antilog(x) = 2^3
Solution:
10^x = 2^3
x = 3log(2) ≈ 0.95424
1. Evaluate: antilog(log(3x))
Solution:
antilog(log(3x)) = 3x
Laws of logarithm
Here are some challenging questions on laws of logarithms:
*Basic Laws*
1. Prove: log(a) + log(b) = log(ab)
1. Simplify: log(2x) + log(3x) using logarithmic properties
1. Evaluate: log(4) / log(2)
*Product Rule*
1. Find log(6) given log(2) = 0.3 and log(3) = 0.47
1. Evaluate: log(10x) given log(10) = 1 and log(x) = 2
1. Simplify: log(abc) using product rule
*Quotient Rule*
1. Find log(8/4) given log(8) = 0.9 and log(4) = 0.6
1. Evaluate: log(x/2) given log(x) = 2 and log(2) = 0.3
1. Simplify: log(a/b) using quotient rule
*Power Rule*
1. Find log(2^3) given log(2) = 0.3
1. Evaluate: log(x^2) given log(x) = 2
1. Simplify: log(a^n) using power rule
*Change of Base*
1. Find log₂(8) given log(8) = 0.9 and log(2) = 0.3
1. Evaluate: log₁₀(x) given log₂(x) = 2 and log₂(10) = 3.32
1. Simplify: log_b(a) using change of base formula
*Applications*
1. Find the pH of a solution with a hydrogen ion concentration of 0.0001 M.
1. Calculate the magnitude of an earthquake with a seismic moment of 10^22
Nm.
1. Determine the number of decibels (dB) for a sound with an intensity of 10^-4
W/m².
*Challenge Problems*
1. Prove: log(a/b) = log(a) - log(b)
1. Solve: x^log(x) = 10
1. Find the value of x: log(2x) = log(x) + log(2)
Solution
Here are the solutions to the challenging questions on laws of logarithms:
_Basic Laws_
1. Prove: log(a) + log(b) = log(ab)
Solution:
By definition of logarithms, log(a) + log(b) = log(a×b) = log(ab)
1. Simplify: log(2x) + log(3x)
Solution:
log(2x) + log(3x) = log(2x×3x) = log(6x²)
1. Evaluate: log(4) / log(2)
Solution:
log(4) / log(2) = log(2²) / log(2) = 2
_Product Rule_
1. Find log(6) given log(2) = 0.3 and log(3) = 0.47
Solution:
log(6) = log(2×3) = log(2) + log(3) = 0.3 + 0.47 = 0.77
1. Evaluate: log(10x) given log(10) = 1 and log(x) = 2
Solution:
log(10x) = log(10) + log(x) = 1 + 2 = 3
1. Simplify: log(abc)
Solution:
log(abc) = log(a) + log(b) + log(c)
_Quotient Rule_
1. Find log(8/4) given log(8) = 0.9 and log(4) = 0.6
Solution:
log(8/4) = log(8) - log(4) = 0.9 - 0.6 = 0.3
1. Evaluate: log(x/2) given log(x) = 2 and log(2) = 0.3
Solution:
log(x/2) = log(x) - log(2) = 2 - 0.3 = 1.7
1. Simplify: log(a/b)
Solution:
log(a/b) = log(a) - log(b)
_Power Rule_
1. Find log(2^3) given log(2) = 0.3
Solution:
log(2^3) = 3log(2) = 3(0.3) = 0.9
1. Evaluate: log(x^2) given log(x) = 2
Solution:
log(x^2) = 2log(x) = 2(2) = 4
1. Simplify: log(a^n)
Solution:
log(a^n) = nlog(a)
_Change of Base_
1. Find log₂(8) given log(8) = 0.9 and log(2) = 0.3
Solution:
log₂(8) = log(8) / log(2) = 0.9 / 0.3 = 3
1. Evaluate: log₁₀(x) given log₂(x) = 2 and log₂(10) = 3.32
Solution:
log₁₀(x) = log₂(x) / log₂(10) = 2 / 3.32 = 0.6024
1. Simplify: log_b(a)
Solution:
log_b(a) = log(a) / log(b)
_Applications_
1. Find the pH of a solution with a hydrogen ion concentration of 0.0001 M.
Solution:
pH = -log(0.0001) = -log(10^(-4)) = 4
1. Calculate the magnitude of an earthquake with a seismic moment of 10^22
Nm.
Solution:
Magnitude = log(10^22) / log(10) = 22
1. Determine the number of decibels (dB) for a sound with an intensity of 10^-4
W/m².
Solution:
dB = 10log(10^-4) = -40 dB
_Challenge Problems_
1. Prove: log(a/b) = log(a) - log(b)
Solution:
By quotient rule, log(a/b) = log(a) - log(b)
1. Solve: x^log(x) = 10
Solution:
Take log both sides, log(x^log(x)) = log(10)
log(x)² = 1
log(x) = ±1
x = 10 or x = 1/10
1. Find the value of x: log(2x) = log(x) + log(2)
Solution:
log(2x) = log(2x)
log(2) + log(x) = log(x) + log(2)
log(2) = log(2)
x=2
Last modified: Sep 27, 2024
