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
2051
Solve for x: log5(x^2) = 2.
Answer:
5 or -5
Step 1: Convert to exponential form: 5^2 = x^2. Step 2: Evaluate the left side: 25 = x^2. Step 3: Solve for x by taking the square root. Since x is squared inside the log, x can be positive or negative. Thus, x = 5 or x = -5.
2052
Simplify: log(8) + log(125).
Answer:
3
Step 1: Use the product rule: log(8) + log(125) = log(8 * 125). Step 2: Multiply the arguments: 8 * 125 = 1000. Step 3: Evaluate log(1000). Since 10^3 = 1000, log(1000) = 3.
2053
If log2(x) = 4, then x^2 equals:
Answer:
256
Step 1: Solve for x by converting to exponential form: x = 2^4. Step 2: Calculate x: x = 16. Step 3: Find x^2: 16^2 = 256.
2054
Evaluate: log2(32) / log2(8).
Answer:
5/3
Step 1: Evaluate the numerator: log2(32) = 5 (since 2^5 = 32). Step 2: Evaluate the denominator: log2(8) = 3 (since 2^3 = 8). Step 3: Divide the results: 5 / 3.
2055
If log(x) = 1.5, what is log(x^2)?
Answer:
3
Step 1: Use the power rule for logarithms: log(x^2) = 2 * log(x). Step 2: Substitute the known value of log(x) into the expression: 2 * 1.5. Step 3: Calculate the product to get 3.
2056
Solve for x: 2^x = 3^(x-1).
Answer:
ln(3) / ln(1.5)
Step 1: Take ln of both sides: x*ln(2) = (x-1)*ln(3). Step 2: Distribute: x*ln(2) = x*ln(3) - ln(3). Step 3: Group x terms: x*ln(3) - x*ln(2) = ln(3), which is x*ln(3/2) = ln(3). Therefore, x = ln(3) / ln(1.5).
2057
Simplify log_a(b) * log_b(a).
Answer:
1
Step 1: Apply the change of base formula to both terms: [ln(b) / ln(a)] * [ln(a) / ln(b)]. Step 2: Notice that the numerators and denominators cancel each other out completely. Step 3: This leaves 1 * 1 = 1. This is a standard identity.
2058
Evaluate 10^(log(5)).
Answer:
5
Step 1: Recognize the base of 'log' is 10. The expression is 10^(log10(5)). Step 2: Use the inverse property a^(log_a(x)) = x. Step 3: Since the bases match (10), the expression evaluates directly to the argument, 5.
2059
What is the base of the logarithm in the expression log(x) if no base is written?
Answer:
10
Step 1: In mathematics, particularly in standard algebra, a logarithm written without a specified base implies the common logarithm. Step 2: The common logarithm uses base 10. Step 3: Therefore, log(x) defaults to log10(x).
2060
Find x if log2(log3(x)) = 1.
Answer:
9
Step 1: Start with the outer logarithm. Convert to exponential form: 2^1 = log3(x), so log3(x) = 2. Step 2: Now solve the inner logarithm by converting to exponential form again: 3^2 = x. Step 3: Calculate 3^2, yielding x = 9.