Logarithms MCQs with Answers & Explanations

Step 1: Use the inverse property of natural logarithms and exponential functions: ln(e^x) = x. Step 2: Substitute this simplification back into the equation. Step 3: We immediately get x = 10.

Step 1: The change of base formula allows calculating a log with an arbitrary base using standard bases. Step 2: The formula states log_b(x) = log_c(x) / log_c(b) for any new base c. Step 3: Using base 10, this is written as log(x) / log(b).

Step 1: Convert from logarithmic form to exponential form: x^2 = 16. Step 2: Take the square root of both sides to solve for x. Step 3: Since the base must be positive, x = 4.

Step 1: Let x = log_4(64). Convert to exponential form: 4^x = 64. Step 2: Identify the power of 4 that yields 64. Step 3: Since 4 * 4 * 4 = 64 (or 4^3 = 64), x = 3.

Step 1: The logarithm has an implied base of 10. Convert to exponential form: 10^1 = x - 2. Step 2: Simplify the equation: 10 = x - 2. Step 3: Add 2 to both sides to solve for x: x = 12.

Step 1: Use the power rule for logarithms to rewrite the exponent: 2*ln(3) = ln(3^2) = ln(9). Step 2: The expression is now e^(ln(9)). Step 3: Apply the inverse property e^(ln(x)) = x. Thus, the result is 9.

Step 1: Rewrite 0.0001 as a power of 10. 0.0001 = 1/10000 = 10^-4. Step 2: The expression is log10(10^-4). Step 3: By the power rule or inverse property, the exponent is the answer: -4.

Step 1: Combine the like terms on the left side: 2*log_3(x) = 4. Step 2: Divide by 2: log_3(x) = 2. Step 3: Convert to exponential form: 3^2 = x. Therefore, x = 9.

Step 1: Evaluate the numerator: log_5(125) = 3 because 5^3 = 125. Step 2: Evaluate the denominator: log_5(25) = 2 because 5^2 = 25. Step 3: Divide the results: 3 / 2.

Step 1: Use the power rule to rewrite the terms: log(x^3) - log(y^2). Step 2: Use the quotient rule for logarithms: log(A) - log(B) = log(A / B). Step 3: Combine them to get log(x^3 / y^2).

Step 1: Express both sides with the same base. Since 1000 = 10^3, write the equation as 10^(x-1) = 10^3. Step 2: Equate the exponents: x - 1 = 3. Step 3: Solve for x by adding 1: x = 4. Alternatively, use logs: log(1000) = x-1 => 3 = x-1 => x=4.

Step 1: Evaluate ln(e), which is 1. The equation becomes ln(x) + 1 = 2. Step 2: Subtract 1 from both sides: ln(x) = 1. Step 3: Convert to exponential form: e^1 = x. Thus, x = e.

Step 1: Use the product rule for logarithms: log(A) + log(B) = log(A * B). Step 2: Multiply the arguments: log(50 * 2) = log(100). Step 3: Evaluate log10(100). Since 10^2 = 100, the answer is 2.

Step-by-step solution: 1. Logarithm asks for the exponent to raise the base to get the argument. 2. 2^5 = 32. 3. Therefore log base 2 of 32 = 5.

Step-by-step solution: 1. Logarithm asks for the exponent to raise the base to get the argument. 2. 3^3 = 27. 3. Therefore log base 3 of 27 = 3.

Step-by-step solution: 1. Logarithm asks for the exponent to raise the base to get the argument. 2. 3^6 = 729. 3. Therefore log base 3 of 729 = 6.

Step-by-step solution: 1. Logarithm asks for the exponent to raise the base to get the argument. 2. 5^3 = 125. 3. Therefore log base 5 of 125 = 3.

Step-by-step solution: 1. Logarithm asks for the exponent to raise the base to get the argument. 2. 4^4 = 256. 3. Therefore log base 4 of 256 = 4.

Step-by-step solution: 1. Logarithm asks for the exponent to raise the base to get the argument. 2. 2^4 = 16. 3. Therefore log base 2 of 16 = 4.

Step-by-step solution: 1. Logarithm asks for the exponent to raise the base to get the argument. 2. 5^4 = 625. 3. Therefore log base 5 of 625 = 4.