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

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a^x, where 'a' is a positive constant and 'x' is the exponent. These functions exhibit rapid growth or decay and are characterized by their base. Understanding how to manipulate and evaluate exponential expressions is crucial for solving problems involving logarithms.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions, expressed as f(x) = log_a(x), where 'a' is the base. They answer the question of what exponent 'b' is needed for a given base 'a' to produce 'x'. Familiarity with properties of logarithms, such as the product, quotient, and power rules, is essential for simplifying and evaluating logarithmic expressions.

Properties of Logarithms

The properties of logarithms include rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule (log_a(bc) = log_a(b) + log_a(c)), the quotient rule (log_a(b/c) = log_a(b) - log_a(c)), and the power rule (log_a(b^c) = c * log_a(b)). These properties are vital for evaluating complex logarithmic expressions and solving equations involving logarithms.