Challenge your understanding of logarithms and exponents with these straightforward questions covering properties, calculations, and real-life applications. Perfect for learners seeking to strengthen their foundations in log rules, exponent operations, and related math concepts.
What is the value of 2 raised to the power of 4?
Explanation: 2 raised to the power of 4 means multiplying 2 by itself four times: 2 x 2 x 2 x 2 = 16. The option '8' is incorrect because 2 to the 3rd power is 8. '12' and '24' are unrelated products and do not result from exponentiation of 2 by 4. Only '16' is the correct answer for this exponential calculation.
What is the value of log₁₀100?
Explanation: The logarithm log₁₀100 asks for the exponent you must put on 10 to get 100, which is 2 because 10² = 100. '10' and '1000' are the base and another power of ten, not the exponent itself. '1' would be the result for log₁₀10, not log₁₀100, so only '2' is correct.
If x = 3, what is the value of 3² × 3³?
Explanation: By the exponent multiplication rule, powers with the same base add their exponents: 3² × 3³ = 3^(2+3) = 3⁵ = 243. '9' is just 3², and '12' is a mistaken addition or multiplication. '243' is the correct value, while the duplicate distractor was provided to test attention.
What is the value of log₂1?
Explanation: Any logarithm of 1 in any base is 0, since any number to the power of 0 is 1. '1' and '-1' are common mistakes when confusing exponent rules. '2' is unrelated here; log₂4 equals 2, not log₂1. Thus, '0' is correct.
What does 5 to the power of -2 equal?
Explanation: 5 to the power of -2 means 1 divided by 5 squared, which is 1/25. The option '-25' incorrectly applies the negative sign to the base. '25' is what you get with a positive exponent. '-1/10' misapplies the negative exponent rule. Only '1/25' expresses the correct inverse.
What is log₅25 equal to?
Explanation: Log₅25 asks for the power to raise 5 to get 25, and since 5² = 25, the answer is 2. '5' would make log₅3125, while '1' is log₅5. '10' is not the right power for the given question. Only '2' matches the exponential relationship.
If y = 2, what is (2³)² equal to?
Explanation: (2³)² means multiplying 2³ by itself, so (2³)² = 2^(3×2) = 2⁶ = 64. '8' is only 2³, not with both exponents. '36' and '12' arise from adding or multiplying base and exponents incorrectly. '64' is the correct application of exponent rules.
What is log₁₀(1000 × 10)?
Explanation: log₁₀(1000 × 10) equals log₁₀1000 plus log₁₀10, which is 3 + 1 = 4 since log(1000) = 3 and log(10) = 1. '3' and '1' miss one of the terms. '5' is overestimating the sum. '4' is the only correct total for the sum of the logarithms.
How is 0.001 expressed as a power of ten?
Explanation: 0.001 is equal to 10 to the power of negative 3 (10⁻³). '10⁻²' would be 0.01, and the positive exponents '10³' and '10²' are 1000 and 100, respectively. Only '10⁻³' accurately represents 0.001.
What is log₅5?
Explanation: The logarithm of a base to itself is always 1 because 5 to the 1st power is 5. '5' confuses the result with the base. '0' would be if the argument was 1. '25' is 5 squared, not the exponent needed for log₅5. Only '1' is correct.