Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given ƒ(x) = 3^x, find ƒ(log₃ 2)

Verified step by step guidance

1. Recognize that the function is given as \(f(x) = 3^x\), and you need to find \(f(\log_3 2)\), which means substituting \(x\) with \(\log_3 2\) in the function.
2. Write the expression explicitly as \(f(\log_3 2) = 3^{\log_3 2}\).
3. Recall the property of exponents and logarithms: for any positive base \(a \neq 1\), \(a^{\log_a b} = b\). This is because the logarithm \(\log_a b\) is the exponent to which \(a\) must be raised to get \(b\).
4. Apply this property to simplify \$3^{\log_3 2}\( directly to \)2$ without further calculation.
5. Conclude that \(f(\log_3 2) = 2\) based on the exponential and logarithmic inverse relationship.

Key Concepts

Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive constant not equal to 1. It models growth or decay processes and has unique properties, such as the function being one-to-one and always positive. Understanding how to evaluate and manipulate these functions is essential for solving problems involving exponents.

Logarithmic Functions

A logarithmic function is the inverse of an exponential function and is written as log_a(x), where a is the base. It answers the question: to what power must the base a be raised to get x? Knowing how to interpret and use logarithms is crucial for simplifying expressions and solving equations involving exponents.

Inverse Function Property of Exponentials and Logarithms

Exponential and logarithmic functions with the same base are inverses, meaning f(log_a(x)) = x and log_a(a^x) = x. This property allows simplification of expressions like f(log_3 2) by directly substituting and canceling the functions, which is key to evaluating the given expression efficiently.