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(2^7)

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, depending on the base. Understanding how to manipulate and evaluate exponential expressions is crucial for solving problems involving logarithms, as they are inversely related.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and are expressed as f(x) = log_a(x), where 'a' is the base. They answer the question: to what exponent must the base 'a' be raised to obtain 'x'? This concept is essential for evaluating expressions involving logarithms, as it allows us to simplify and solve equations involving exponential growth.

Properties of Logarithms

The properties of logarithms, such as the product, quotient, and power rules, provide tools for simplifying logarithmic expressions. For example, log_a(b * c) = log_a(b) + log_a(c) and log_a(b^c) = c * log_a(b). These properties are vital for evaluating logarithmic functions efficiently, especially when dealing with complex expressions or changing bases.