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
2061
Express 3ln(2) - ln(4) as a single natural logarithm.
Answer:
ln(2)
Step 1: Use the power rule to rewrite 3ln(2) as ln(2^3), which is ln(8). Step 2: The expression is now ln(8) - ln(4). Step 3: Use the quotient rule: ln(8) - ln(4) = ln(8/4) = ln(2).
2062
Find the value of log(1000^2).
Answer:
6
Step 1: Use the power rule to bring the exponent to the front: 2 * log(1000). Step 2: Evaluate log(1000). Since 10^3 = 1000, log(1000) = 3. Step 3: Multiply the results: 2 * 3 = 6.
2063
Solve for x: log(x+3) = log(2x).
Answer:
3
Step 1: Because the logarithms have the same base and are equal, their arguments must be equal: x + 3 = 2x. Step 2: Subtract x from both sides to isolate x. Step 3: This gives 3 = x, so x = 3.
2064
Evaluate: log2(1/2) + log3(1/3).
Answer:
-2
Step 1: Find log2(1/2). Since 2^-1 = 1/2, log2(1/2) = -1. Step 2: Find log3(1/3). Since 3^-1 = 1/3, log3(1/3) = -1. Step 3: Add the two results: -1 + (-1) = -2.
2065
Convert the equation 5^3 = 125 into logarithmic form.
Answer:
log5(125) = 3
Step 1: The basic conversion rule is a^b = c is equivalent to log_a(c) = b. Step 2: Identify the base a=5, exponent b=3, and result c=125. Step 3: Substitute these into the logarithmic form to get log5(125) = 3.
2066
What is the domain of the function f(x) = log(x - 5)?
Answer:
x > 5
Step 1: The argument of a logarithm must be strictly greater than zero. Step 2: Set the argument greater than zero: x - 5 > 0. Step 3: Solve the inequality to get x > 5. Thus, the domain is all real numbers strictly greater than 5.
2067
Solve for x: e^(2x) = 20.
Answer:
ln(20) / 2
Step 1: To isolate the exponent, take the natural logarithm (ln) of both sides: ln(e^(2x)) = ln(20). Step 2: Using the property ln(e^y) = y, simplify the left side to 2x = ln(20). Step 3: Divide by 2 to solve for x: x = ln(20) / 2.
2068
If log(a) = 4 and log(b) = 1, find log(a/b).
Answer:
3
Step 1: Use the quotient rule for logarithms: log(a/b) = log(a) - log(b). Step 2: Substitute the given values into the equation: log(a/b) = 4 - 1. Step 3: Perform the subtraction to get 3.
2069
If log(x) = 3 and log(y) = 2, find log(xy).
Answer:
5
Step 1: Use the product rule for logarithms: log(xy) = log(x) + log(y). Step 2: Substitute the given values into the equation: log(xy) = 3 + 2. Step 3: Perform the addition to get 5.
2070
Which statement is false?
Answer:
log(A+B) = log(A) + log(B)
Step 1: Review the standard properties of logarithms. Options A, B, and D correspond to the product, quotient, and power rules. Step 2: Check Option C: log(A+B). There is no logarithmic property that distributes a log over addition. Step 3: Therefore, log(A+B) = log(A) + log(B) is a false statement.