All Categories MCQs
Topic Notes: All Categories
General Description
Plato
- Biography: Ancient Greek philosopher (427–347 BCE), student of Socrates and teacher of Aristotle, founder of the Academy in Athens.
- Important Ideas:
- Theory of Forms
- Philosopher-King
- Ideal State
2071
Find the value of log_sqrt(3)(9).
Answer:
4
Step 1: Let x = log_sqrt(3)(9). Convert to exponential form: (sqrt(3))^x = 9. Step 2: Write both sides with base 3: (3^(1/2))^x = 3^2, which becomes 3^(x/2) = 3^2. Step 3: Equate exponents: x/2 = 2. Multiply by 2 to get x = 4.
2072
Simplify log(a^2 * b^3).
Answer:
2log(a) + 3log(b)
Step 1: Apply the product rule: log(A*B) = log(A) + log(B). This gives log(a^2) + log(b^3). Step 2: Apply the power rule to both terms: log(x^k) = k*log(x). Step 3: This transforms the expression to 2*log(a) + 3*log(b).
2073
Evaluate log8(2).
Answer:
1/3
Step 1: Let x = log8(2). Convert to exponential form: 8^x = 2. Step 2: Express base 8 as 2^3. This gives (2^3)^x = 2^1, or 2^(3x) = 2^1. Step 3: Equate the exponents: 3x = 1, so x = 1/3.
2074
What is the value of log2(0)?
Answer:
Undefined
Step 1: The logarithm log_b(x) is only defined for positive real numbers (x > 0). Step 2: The domain of any logarithmic function does not include 0 or negative numbers. Step 3: Therefore, log2(0) is mathematically undefined (though it approaches negative infinity as a limit).
2075
Solve for x: log3(x) + log3(x-2) = 1.
Answer:
3
Step 1: Combine using product rule: log3(x(x-2)) = 1. Step 2: Convert to exponential: x^2 - 2x = 3^1, or x^2 - 2x - 3 = 0. Step 3: Factor: (x-3)(x+1) = 0, so x=3 or x=-1. Since log argument must be positive, x=-1 is invalid. Thus, x=3.
2076
Which of the following is equivalent to -log(x)?
Answer:
log(1/x)
Step 1: Use the power rule in reverse: k * log(x) = log(x^k). Step 2: Here, k = -1, so -log(x) = -1 * log(x) = log(x^-1). Step 3: Rewrite the negative exponent as a fraction: log(1/x).
2077
Solve for x: ln(x) = 0.
Answer:
1
Step 1: The natural logarithm ln has base e. The equation is log_e(x) = 0. Step 2: Convert to exponential form: e^0 = x. Step 3: Any non-zero number to the power of 0 is 1. Therefore, x = 1.
2078
Find the value of log(sqrt(10)).
Answer:
0.5
Step 1: Write the square root as a fractional exponent: sqrt(10) = 10^(1/2). Step 2: The expression becomes log10(10^(1/2)). Step 3: By the power rule, this equals (1/2) * log10(10). Since log10(10) = 1, the answer is 1/2 or 0.5.
2079
Evaluate 2^log2(7).
Answer:
7
Step 1: Recognize the fundamental identity of logarithms: a^(log_a(x)) = x. Step 2: In this expression, the base of the exponent (2) is the same as the base of the logarithm (2). Step 3: Therefore, the expression simplifies directly to the argument, which is 7.
2080
Simplify: log(50) + log(20).
Answer:
3
Step 1: Apply the product rule: log(A) + log(B) = log(A * B). Step 2: log(50) + log(20) = log(50 * 20) = log(1000). Step 3: Since 1000 = 10^3, log(1000) = 3.