Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given ƒ(x) = log₂ x, find ƒ(2⁷)

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Identify the function given: \( f(x) = \log_2 x \), which means the logarithm base 2 of \( x \).

Substitute the input \( 2^7 \) into the function: \( f(2^7) = \log_2 (2^7) \).

Recall the logarithmic property that \( \log_b (b^k) = k \), where \( b \) is the base of the logarithm and \( k \) is the exponent.

Apply this property to simplify \( \log_2 (2^7) \) to just the exponent \( 7 \).

Conclude that \( f(2^7) = 7 \) based on the simplification.

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주요 개념

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Exponential Functions

Exponential functions have the form f(x) = a^x, where the variable is in the exponent. Understanding how to manipulate and evaluate expressions like 2^7 is essential, as these functions grow rapidly and are the inverse of logarithmic functions.

Logarithmic Functions

A logarithmic function, such as f(x) = log_2 x, is the inverse of an exponential function. It answers the question: 'To what power must the base 2 be raised to get x?' Recognizing this inverse relationship helps simplify expressions like f(2^7).

Properties of Logarithms and Exponents

Key properties include log_b(b^x) = x and b^{log_b x} = x. These allow simplification of expressions involving logs and exponents by 'canceling' the operations when the base matches, which is crucial for evaluating f(2^7) when f(x) = log_2 x.

Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given ƒ(x) = log2 x, find ƒ(27)