Logarithms MCQs with Answers & Explanations (Page 3)

Step 1: Recognize the fundamental identity of logarithms: a^(log_a(x)) = x. Step 2: In this expression, the base of the exponent (2) is the same as the base of the logarithm (2). Step 3: Therefore, the expression simplifies directly to the argument, which is 7.

Step 1: Write the square root as a fractional exponent: sqrt(10) = 10^(1/2). Step 2: The expression becomes log10(10^(1/2)). Step 3: By the power rule, this equals (1/2) * log10(10). Since log10(10) = 1, the answer is 1/2 or 0.5.

Step 1: The natural logarithm ln has base e. The equation is log_e(x) = 0. Step 2: Convert to exponential form: e^0 = x. Step 3: Any non-zero number to the power of 0 is 1. Therefore, x = 1.

Step 1: Use the power rule in reverse: k * log(x) = log(x^k). Step 2: Here, k = -1, so -log(x) = -1 * log(x) = log(x^-1). Step 3: Rewrite the negative exponent as a fraction: log(1/x).

Step 1: Combine using product rule: log3(x(x-2)) = 1. Step 2: Convert to exponential: x^2 - 2x = 3^1, or x^2 - 2x - 3 = 0. Step 3: Factor: (x-3)(x+1) = 0, so x=3 or x=-1. Since log argument must be positive, x=-1 is invalid. Thus, x=3.

Step 1: The logarithm log_b(x) is only defined for positive real numbers (x > 0). Step 2: The domain of any logarithmic function does not include 0 or negative numbers. Step 3: Therefore, log2(0) is mathematically undefined (though it approaches negative infinity as a limit).

Step 1: Let x = log8(2). Convert to exponential form: 8^x = 2. Step 2: Express base 8 as 2^3. This gives (2^3)^x = 2^1, or 2^(3x) = 2^1. Step 3: Equate the exponents: 3x = 1, so x = 1/3.

Step 1: Apply the product rule: log(A*B) = log(A) + log(B). This gives log(a^2) + log(b^3). Step 2: Apply the power rule to both terms: log(x^k) = k*log(x). Step 3: This transforms the expression to 2*log(a) + 3*log(b).

Step 1: Let x = log_sqrt(3)(9). Convert to exponential form: (sqrt(3))^x = 9. Step 2: Write both sides with base 3: (3^(1/2))^x = 3^2, which becomes 3^(x/2) = 3^2. Step 3: Equate exponents: x/2 = 2. Multiply by 2 to get x = 4.

Step 1: Review the standard properties of logarithms. Options A, B, and D correspond to the product, quotient, and power rules. Step 2: Check Option C: log(A+B). There is no logarithmic property that distributes a log over addition. Step 3: Therefore, log(A+B) = log(A) + log(B) is a false statement.

Step 1: Use the product rule for logarithms: log(xy) = log(x) + log(y). Step 2: Substitute the given values into the equation: log(xy) = 3 + 2. Step 3: Perform the addition to get 5.

Step 1: Use the quotient rule for logarithms: log(a/b) = log(a) - log(b). Step 2: Substitute the given values into the equation: log(a/b) = 4 - 1. Step 3: Perform the subtraction to get 3.

Step 1: To isolate the exponent, take the natural logarithm (ln) of both sides: ln(e^(2x)) = ln(20). Step 2: Using the property ln(e^y) = y, simplify the left side to 2x = ln(20). Step 3: Divide by 2 to solve for x: x = ln(20) / 2.

Step 1: The argument of a logarithm must be strictly greater than zero. Step 2: Set the argument greater than zero: x - 5 > 0. Step 3: Solve the inequality to get x > 5. Thus, the domain is all real numbers strictly greater than 5.

Step 1: The basic conversion rule is a^b = c is equivalent to log_a(c) = b. Step 2: Identify the base a=5, exponent b=3, and result c=125. Step 3: Substitute these into the logarithmic form to get log5(125) = 3.

Step 1: Find log2(1/2). Since 2^-1 = 1/2, log2(1/2) = -1. Step 2: Find log3(1/3). Since 3^-1 = 1/3, log3(1/3) = -1. Step 3: Add the two results: -1 + (-1) = -2.

Step 1: Because the logarithms have the same base and are equal, their arguments must be equal: x + 3 = 2x. Step 2: Subtract x from both sides to isolate x. Step 3: This gives 3 = x, so x = 3.

Step 1: Use the power rule to bring the exponent to the front: 2 * log(1000). Step 2: Evaluate log(1000). Since 10^3 = 1000, log(1000) = 3. Step 3: Multiply the results: 2 * 3 = 6.

Step 1: Use the power rule to rewrite 3ln(2) as ln(2^3), which is ln(8). Step 2: The expression is now ln(8) - ln(4). Step 3: Use the quotient rule: ln(8) - ln(4) = ln(8/4) = ln(2).

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.