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
2031
Calculate log_10(0.001).
Answer:
-3
Step 1: Let x = log10(0.001). Convert to exponential form: 10^x = 0.001. Step 2: Express 0.001 as a fraction: 1/1000, which is 10^-3. Step 3: Equating exponents in 10^x = 10^-3 gives x = -3.
2032
If ln(x) = y, then what is e^y?
Answer:
x
Step 1: By definition, ln(x) means log_e(x). So log_e(x) = y. Step 2: Convert this logarithmic equation into its exponential form. Step 3: The base e raised to the power of y equals x (e^y = x).
2033
Solve for x: log2(x-1) = 3.
Answer:
9
Step 1: Convert the logarithmic equation to its exponential form: 2^3 = x - 1. Step 2: Evaluate 2^3 to get 8, so 8 = x - 1. Step 3: Add 1 to both sides to solve for x, resulting in x = 9.
2034
Find the value of log(25) + log(4).
Answer:
2
Step 1: Use the product property of logarithms: log(A) + log(B) = log(A * B). Step 2: Multiply the arguments: log(25 * 4) = log(100). Step 3: Since 10^2 = 100, log(100) evaluates to 2.
2035
Express 1/2 * log(x) as a single logarithm.
Answer:
log(sqrt(x))
Step 1: Use the power rule for logarithms backwards: k * log(A) = log(A^k). Step 2: Apply this to the expression: log(x^(1/2)). Step 3: Since a fractional exponent of 1/2 represents the square root, this equals log(sqrt(x)).
2036
Solve for x: e^(ln(x+2)) = 7.
Answer:
5
Step 1: Apply the inverse property e^(ln(A)) = A to the left side. Step 2: The equation simplifies to x + 2 = 7. Step 3: Subtract 2 from both sides to find x = 5.
2037
What is the equivalent of log_2(x) / log_2(10)?
Answer:
log_10(x)
Step 1: Recognize the change of base formula in reverse: log_c(a) / log_c(b) = log_b(a). Step 2: Here, c = 2, a = x, and b = 10. Step 3: Substituting these values into the formula yields log_10(x), which is the common log(x).
2038
If log_a(2) = x and log_a(3) = y, write log_a(12) in terms of x and y.
Answer:
2x + y
Step 1: Prime factorize 12: 12 = 4 * 3 = 2^2 * 3. Step 2: Apply logarithm rules: log_a(2^2 * 3) = log_a(2^2) + log_a(3) = 2*log_a(2) + log_a(3). Step 3: Substitute x and y: 2(x) + y = 2x + y.
2039
Solve for x: log5(x) = -2.
Answer:
0.04
Step 1: Rewrite the equation in exponential form: 5^-2 = x. Step 2: A negative exponent means reciprocal: x = 1 / (5^2). Step 3: Calculate 5^2 to get 25. Thus, x = 1/25 = 0.04.
2040
Evaluate log_x(x).
Answer:
1
Step 1: The logarithm asks 'to what power must we raise the base x to get the argument x?'. Step 2: Since x^1 = x for any valid base x. Step 3: Therefore, log_x(x) = 1.