Answer each of the following. Between what two consecutive integers must log₂12 lie?

검증된 단계별 안내

Recall that \( \log_2 12 \) means the exponent to which 2 must be raised to get 12.

Identify powers of 2 that are close to 12. For example, \( 2^3 = 8 \) and \( 2^4 = 16 \).

Since 12 is between 8 and 16, \( \log_2 12 \) must be between 3 and 4 because \( 2^3 < 12 < 2^4 \).

Therefore, the two consecutive integers between which \( \log_2 12 \) lies are 3 and 4.

This means \( 3 < \log_2 12 < 4 \).

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Logarithms and Their Meaning

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log₂12 asks for the power to which 2 must be raised to get 12. Understanding this helps in estimating or calculating logarithmic values.

Properties of Logarithms

Logarithms have properties such as log_b(mn) = log_b(m) + log_b(n), which can simplify calculations. Recognizing that 12 lies between powers of 2 (like 8 and 16) helps estimate log₂12 by comparing it to known integer logarithms.

Estimating Logarithmic Values Using Consecutive Integers

To find between which two integers a logarithm lies, identify consecutive powers of the base that bracket the number. Since 2³ = 8 and 2⁴ = 16, and 12 is between 8 and 16, log₂12 must lie between 3 and 4.

Answer each of the following. Between what two consecutive integers must log2 12 lie?