Logarithms MCQs with Answers & Explanations (Page 4)

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).

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.

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.

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).

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.

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.

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.

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.

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.

Step 1: Recall the change of base formula: log_b(a) = ln(a) / ln(b). Step 2: Recall that log_a(b) = ln(b) / ln(a). Step 3: Therefore, log_b(a) is the reciprocal of log_a(b), written as 1 / log_a(b).

Step 1: Let x = log3(1/27). In exponential form, 3^x = 1/27. Step 2: Express 1/27 as a base of 3: 1/27 = 1/(3^3) = 3^-3. Step 3: Equate the exponents in 3^x = 3^-3 to get x = -3.

Step 1: The natural logarithm 'ln' has base 'e'. Step 2: Convert the equation log_e(x) = 2.5 into exponential form. Step 3: The base 'e' raised to the exponent 2.5 equals the argument x. Thus, x = e^2.5.

Step 1: Convert to exponential form: 4^1.5 = x. Step 2: Write 1.5 as 3/2. So, x = 4^(3/2). Step 3: This means the square root of 4, cubed. sqrt(4) = 2, and 2^3 = 8. Thus, x = 8.

Step 1: Let x = log_sqrt(2)(8). Convert to exponential form: (sqrt(2))^x = 8. Step 2: Express both sides with base 2: (2^(1/2))^x = 2^3, which is 2^(x/2) = 2^3. Step 3: Equate the exponents: x/2 = 3, yielding x = 6.

Step 1: Use the quotient rule to split the fraction: ln(x^2) - ln(y). Step 2: Use the power rule on the first term: 2*ln(x). Step 3: Combine them to get the final expanded form: 2*ln(x) - ln(y).

Step 1: Expand using the product rule: log(10 * x * y) = log(10) + log(x) + log(y). Step 2: Evaluate log(10) (base 10 is implied), which is 1. Step 3: Substitute a and b for log(x) and log(y) to get 1 + a + b.

Step 1: Combine the left side using the product rule: log((x+1)(x-1)) = log(3). Step 2: Equate the arguments: x^2 - 1 = 3. Step 3: x^2 = 4, so x = 2 or x = -2. Since log argument must be positive, x = -2 is invalid. Thus, x = 2.

Step 1: Use the power rule on the exponent: 2log(3) becomes log(3^2) = log(9). Step 2: The expression is now 10^(log(9)). Step 3: Apply the inverse property 10^(log(x)) = x. Therefore, the result is 9.

Step 1: Let x = log2(1024). Convert to exponential form: 2^x = 1024. Step 2: Determine the power of 2 that equals 1024. 2^5 = 32, 2^10 = 1024. Step 3: Equating exponents gives x = 10.

Step 1: The logarithm asks 'to what power must we raise the base x to get the argument x?'. Step 2: Since x^1 = x for any valid base x. Step 3: Therefore, log_x(x) = 1.