Logarithms MCQs with Answers & Explanations

Step 1: Let x = log2(8). Step 2: Convert the logarithmic equation to its exponential form: 2^x = 8. Step 3: Since 8 can be written as 2^3, we have 2^x = 2^3. Therefore, x = 3.

Step 1: Set the expression equal to x: log3(81) = x. Step 2: Rewrite in exponential form: 3^x = 81. Step 3: Recognize that 3^4 = 81. Thus, 3^x = 3^4, which means x = 4.

Step 1: Write the equation log10(1000) = x. Step 2: Convert to exponential form: 10^x = 1000. Step 3: Since 1000 is 10 cubed (10^3), we get 10^x = 10^3, so x = 3.

Step 1: Use the identity log_a(a) = 1. Step 2: Let log7(7) = x. In exponential form, 7^x = 7. Step 3: Since 7^1 = 7, it follows that x = 1.

Step 1: Let x = log2(1/16). Convert to exponential form: 2^x = 1/16. Step 2: Express 1/16 as a power of 2: 1/16 = 1/(2^4) = 2^-4. Step 3: Equating the exponents in 2^x = 2^-4 gives x = -4.

Step 1: Let x = log10(0.01). Convert to exponential form: 10^x = 0.01. Step 2: Write 0.01 as a fraction: 1/100, which is 10^-2. Step 3: So, 10^x = 10^-2, which implies x = -2.

Step 1: Let log4(2) = x. Convert to exponential form: 4^x = 2. Step 2: Express both sides with the same base (base 2). Since 4 = 2^2, we have (2^2)^x = 2^1, or 2^(2x) = 2^1. Step 3: Equate the exponents: 2x = 1, so x = 1/2.

Step 1: Let x = log9(3). In exponential form, 9^x = 3. Step 2: Express base 9 as 3^2. This gives (3^2)^x = 3^1, meaning 3^(2x) = 3^1. Step 3: Equating exponents gives 2x = 1, resulting in x = 1/2.

Step 1: Let x = log(1/2)(8). Exponential form is (1/2)^x = 8. Step 2: Rewrite both sides using base 2. (1/2) is 2^-1, and 8 is 2^3. Step 3: This gives (2^-1)^x = 2^3, so 2^(-x) = 2^3. Therefore, -x = 3, which means x = -3.

Step 1: Evaluate each logarithm separately. log2(64) = 6 because 2^6 = 64. Step 2: log2(4) = 2 because 2^2 = 4. Step 3: Add the results: 6 + 2 = 8. Alternatively, use the product rule: log2(64 * 4) = log2(256) = 8.

Step 1: Evaluate each term. log3(27) = 3 (since 3^3 = 27) and log3(9) = 2 (since 3^2 = 9). Step 2: Subtract the values: 3 - 2 = 1. Step 3: Alternatively, use the quotient rule: log3(27/9) = log3(3) = 1.

Step 1: Use the product rule for logarithms: log_b(M) + log_b(N) = log_b(M * N). Step 2: Apply the rule: log10(5) + log10(2) = log10(5 * 2) = log10(10). Step 3: Since log_b(b) = 1, log10(10) = 1.

Step 1: Use the quotient rule for logarithms: log_b(M) - log_b(N) = log_b(M / N). Step 2: Apply the rule: log6(72) - log6(2) = log6(72 / 2) = log6(36). Step 3: Evaluate log6(36). Since 6^2 = 36, the value is 2.

Step 1: Recognize the value of log2(2). Since the base and argument are the same, log2(2) = 1. Step 2: Substitute this value back into the expression: 3 * 1. Step 3: The result is 3.

Step 1: Recall the power rule for logarithms, which states that log_b(M^k) = k * log_b(M). Step 2: Apply this rule to the expression log(x^3). Step 3: The exponent 3 moves to the front, resulting in 3 * log(x).

Step 1: A logarithm written without a base implies base 10 (common logarithm). So, log10(x) = 2. Step 2: Convert to exponential form: 10^2 = x. Step 3: Calculate 10^2, which equals 100. Therefore, x = 100.

Step 1: Identify the base (2), the exponent (5), and the argument (x). Step 2: Convert the logarithmic equation to exponential form: 2^5 = x. Step 3: Calculate 2^5: 2 * 2 * 2 * 2 * 2 = 32. Thus, x = 32.

Step 1: Convert the equation from logarithmic to exponential form: 3^2 = x - 1. Step 2: Simplify the exponential term: 9 = x - 1. Step 3: Add 1 to both sides to solve for x: x = 9 + 1 = 10.

Step 1: Convert to exponential form: 5^2 = 2x + 1. Step 2: Simplify the left side: 25 = 2x + 1. Step 3: Subtract 1 and divide by 2: 24 = 2x, so x = 12.

Step 1: Convert the logarithmic equation to exponential form: x^4 = 81. Step 2: Take the fourth root of both sides. Step 3: Since 3^4 = 81 (and the base of a logarithm must be positive), x = 3.