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^{2 log_2 2})

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Identify the given function and expression: ƒ(x) = log_2 x, and we need to find ƒ(2^{2 \(\log\)_2 2}).

Recall that ƒ(x) = log_2 x means the function outputs the logarithm base 2 of its input, so ƒ(2^{2 \(\log\)_2 2}) = \(\log\)_2 \(\left\)(2^{2 \(\log\)_2 2}\(\right\)).

Use the logarithmic property \(\log\)_b (a^c) = c \(\log\)_b a to simplify the expression inside the logarithm: \(\log\)_2 \(\left\)(2^{2 \(\log\)_2 2}\(\right\)) = 2 \(\log\)_2 2 \(\times\) \(\log\)_2 2.

Recognize that \(\log\)_2 2 equals 1 because 2 raised to the power 1 is 2.

Substitute \(\log\)_2 2 = 1 back into the expression and simplify the multiplication to find the value of the function at the given input.

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Logarithmic Functions and Their Properties

A logarithmic function log_b(x) is the inverse of the exponential function b^x. It satisfies properties such as log_b(b^x) = x and log_b(xy) = log_b(x) + log_b(y), which help simplify expressions involving logs and exponents.

Exponential Functions and Their Properties

An exponential function has the form b^x, where b is a positive base not equal to 1. Key properties include b^{m+n} = b^m * b^n and (b^m)^n = b^{mn}, which allow manipulation and simplification of expressions with exponents.

Inverse Relationship Between Logarithms and Exponents

Logarithms and exponents are inverse operations, meaning log_b(b^x) = x and b^{log_b(x)} = x. This relationship is essential for simplifying expressions where logs and exponents are nested, as in the given function evaluation.

Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given ƒ(x) = log2 x, find ƒ(22 log_2 2)