Logarithms MCQs with Answers & Explanations

Step 1: Rewrite the equation in exponential form: x^3 = 125. Step 2: Take the cube root of both sides. Step 3: Since 5 * 5 * 5 = 125, the cube root of 125 is 5. Therefore, x = 5.

Step 1: Logarithms can have any positive base except 1. Step 2: The common logarithm 'log' defaults to base 10. Step 3: The natural logarithm 'ln' is mathematically defined as having the base 'e', Euler's number (approximately 2.718).

Step 1: Recall that 'ln' is logarithm base e. So the expression is log_e(e^5). Step 2: Use the power rule for logarithms: ln(e^5) = 5 * ln(e). Step 3: Since ln(e) = 1, the expression simplifies to 5 * 1 = 5.

Step 1: Use the inverse property of exponential and logarithmic functions: a^(log_a(x)) = x. Step 2: Here, the base of the exponential is 'e', and the base of 'ln' is also 'e'. Step 3: Applying the property gives e^(ln(7)) = 7.

Step 1: The change of base formula is log_b(a) = log_c(a) / log_c(b). Step 2: To change to common logarithms (base 10), we set c = 10. Step 3: Substituting a = 10 and b = 2 gives log_10(10) / log_10(2), which is written as log(10) / log(2).

Step 1: Use the change of base formula: log_b(a) = ln(a)/ln(b). Step 2: Rewrite the expression: [ln(3)/ln(2)] * [ln(4)/ln(3)]. Step 3: Cancel out ln(3) to get ln(4)/ln(2), which is log2(4). Since 2^2 = 4, the answer is 2.

Step 1: Use the power rule to bring the exponent to the front: x * log5(25). Step 2: Evaluate log5(25). Since 5^2 = 25, log5(25) = 2. Step 3: Multiply the result by x to get x * 2, which is 2x.

Step 1: Use the power rule to rewrite 2 * log(3) as log(3^2), which is log(9). Step 2: The expression becomes log(9) + log(4). Step 3: Use the product rule: log(A) + log(B) = log(A * B). This gives log(9 * 4) = log(36).

Step 1: Use the product rule log(AB) = log(A) + log(B) to split the term: log(x^2) + log(y). Step 2: Use the power rule log(A^k) = k * log(A) on the first term. Step 3: log(x^2) becomes 2 * log(x). The final expression is 2 * log(x) + log(y).

Step 1: Use the product rule on the left side: log(x * 2) = log(14), so log(2x) = log(14). Step 2: Since the logarithmic function is one-to-one, we can equate the arguments: 2x = 14. Step 3: Divide by 2 to solve for x: x = 14 / 2 = 7.

Step 1: Use the quotient rule to combine the logarithms: log2(x / 3) = 2. Step 2: Convert to exponential form: 2^2 = x / 3. Step 3: Simplify to 4 = x / 3, then multiply by 3 to get x = 12.

Step 1: Take the natural logarithm (ln) of both sides: ln(3^x) = ln(15). Step 2: Use the power rule to bring x down: x * ln(3) = ln(15). Step 3: Divide both sides by ln(3) to isolate x: x = ln(15) / ln(3).

Step 1: The equation is in base 10. Take the common logarithm (log base 10) of both sides: log(10^x) = log(40). Step 2: Using the power rule, x * log(10) = log(40). Step 3: Since log(10) = 1, the equation simplifies exactly to x = log(40).

Step 1: Evaluate log(100) which is log_10(10^2) = 2. Step 2: Evaluate log(0.1) which is log_10(10^-1) = -1. Step 3: Add the results: 2 + (-1) = 1.

Step 1: The characteristic of a common logarithm is the integer part. Step 2: For a number N > 1, the characteristic is one less than the number of digits to the left of the decimal point. Step 3: 456.7 has 3 digits before the decimal. The characteristic is 3 - 1 = 2.

Step 1: For a number N < 1, the characteristic is negative. Step 2: Count the number of zeros immediately after the decimal point and add 1. Make it negative. Step 3: 0.0073 has two zeros after the decimal point. So, -(2 + 1) = -3.

Step 1: Write 8 as a power of 2: 8 = 2^3. Step 2: Substitute this into the logarithm: log(8) = log(2^3) = 3 * log(2). Step 3: Multiply the given value: 3 * 0.3010 = 0.9030.

Step 1: Express 300 in scientific notation or factor it: 300 = 3 * 100. Step 2: Use the product rule: log(300) = log(3) + log(100). Step 3: Substitute known values: 0.4771 + 2 = 2.4771.

Step 1: The mantissa must always be positive. -1.5 = -2 + 0.5. Step 2: The characteristic is -2, often written as 2_bar. Step 3: The mantissa is 0.5000. So the representation is 2_bar.5000.

Step 1: Apply the product rule: log(A) + log(B) = log(A * B). Step 2: log(50) + log(20) = log(50 * 20) = log(1000). Step 3: Since 1000 = 10^3, log(1000) = 3.