Logarithms MCQs with Answers & Explanations (Page 5)

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.

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.

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).

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.

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)).

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.

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.

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).

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.

Step 1: Convert the logarithmic expression into exponential form: x^2 = 64. Step 2: Take the square root of both sides to solve for x. Step 3: Since x must be a positive base, x = 8.

Step 1: Review the quotient rule for logarithms. Step 2: The quotient rule states that the logarithm of a division is the difference of the logarithms. Step 3: Therefore, log(A / B) translates exactly to log(A) - log(B).

Step 1: Use the power rule to bring the exponent to the front: 7 * log5(5). Step 2: Use the identity log_a(a) = 1. So, log5(5) = 1. Step 3: Multiply the values: 7 * 1 = 7. Alternatively, use the inverse property log_a(a^x) = x.

Step 1: Convert from logarithmic to exponential form: 3^-1 = x. Step 2: A negative exponent signifies the reciprocal of the base. Step 3: Therefore, x = 1 / 3^1 = 1/3.

Step 1: Use the quotient property of logarithms: log(A) - log(B) = log(A / B). Step 2: Apply the property: log(200 / 2) = log(100). Step 3: Since 10^2 = 100, log(100) = 2.

Step 1: Apply the product rule: log(A*B) = log(A) + log(B), which gives log(x^3) + log(y^2). Step 2: Apply the power rule to each term: log(x^k) = k*log(x). Step 3: This transforms log(x^3) to 3*log(x) and log(y^2) to 2*log(y). Total is 3*log(x) + 2*log(y).

Step 1: Convert to exponential form: 2^4 = 2x. Step 2: Evaluate 2^4 to get 16. The equation is 16 = 2x. Step 3: Divide both sides by 2 to solve for x, giving x = 8.

Step 1: This expression asks 'to what power must base a be raised to equal a^x?'. Step 2: The power is clearly x. Step 3: Alternatively, use the power rule: x * log_a(a) = x * 1 = x.

Step 1: Express 50 as a product of 5 and 10: log(50) = log(5 * 10). Step 2: Use the product rule: log(5) + log(10). Step 3: Substitute known values: 0.699 + 1 = 1.699.

Step 1: Use the power rule to move the negative sign: -log(2) = log(2^-1) = log(1/2). Step 2: The expression becomes 10^(log(1/2)). Step 3: Apply the inverse property 10^(log(x)) = x. The result is 1/2 or 0.5.

Step 1: Let x = log2(1/32). Convert to exponential form: 2^x = 1/32. Step 2: Express 1/32 as a base of 2. Since 32 = 2^5, 1/32 = 2^-5. Step 3: Equate exponents: 2^x = 2^-5, giving x = -5.