Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given ƒ(x) = log₂ x, find ƒ(2^{log_2 2})

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Identify the given function and the expression to evaluate: ƒ(x) = log_2 x, and we need to find ƒ(2^{log_2 2}).

Recall that ƒ(x) = log_2 x means the logarithm is base 2, so ƒ(2^{log_2 2}) = log_2 (2^{log_2 2}).

Use the logarithmic property that \( \log_b (b^k) = k \) to simplify the expression inside the logarithm.

Apply this property: \( \log_2 (2^{log_2 2}) = log_2 2 \), because the logarithm and the exponent base are the same.

Evaluate \( log_2 2 \) by recognizing that 2 is the base of the logarithm, so \( log_2 2 = 1 \).

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Logarithmic Functions

A logarithmic function is the inverse of an exponential function. For a base b > 0 and b ≠ 1, log_b(x) answers the question: to what power must b be raised to get x? Understanding how to evaluate and manipulate logarithms is essential for solving expressions involving log functions.

Exponential Functions

An exponential function has the form b^x, where the base b is a positive constant not equal to 1. These functions grow or decay rapidly and are the inverse operations of logarithms. Recognizing how to simplify expressions like 2^(log_2 2) relies on understanding this inverse relationship.

Inverse Properties of Logarithms and Exponentials

Logarithmic and exponential functions with the same base are inverses, meaning log_b(b^x) = x and b^(log_b x) = x. This property allows simplification of nested expressions, such as f(2^(log_2 2)), by 'canceling' the log and exponential when bases match.

Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given ƒ(x) = log2 x, find ƒ(2log_2 2)