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})

Verified step by step guidance

1. Identify the given function and expression: ƒ(x) = log_2 x, and we need to find ƒ(2^{2 \log_2 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).
3. 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.
4. Recognize that \log_2 2 equals 1 because 2 raised to the power 1 is 2.
5. Substitute \log_2 2 = 1 back into the expression and simplify the multiplication to find the value of the function at the given input.

Key Concepts

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.