Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given ƒ(x) = 3^x, find ƒ(log_3 (ln 3))

Verified step by step guidance

1. Identify the function given: \( f(x) = 3^x \).
2. Recognize that you need to evaluate \( f(\log_3(\ln 3)) \).
3. Understand that \( \log_3(\ln 3) \) is the exponent to which 3 must be raised to get \( \ln 3 \).
4. Apply the property of logarithms: \( 3^{\log_3(\ln 3)} = \ln 3 \).
5. Conclude that \( f(\log_3(\ln 3)) = \ln 3 \) by substituting back into the function \( f(x) = 3^x \).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form ƒ(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. In this context, ƒ(x) = 3^x represents an exponential function with base 3, which is crucial for evaluating expressions involving logarithms.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and are expressed as log_b(a) = c, meaning b^c = a. They help in solving equations where the variable is an exponent. In the given problem, log_3(ln 3) indicates the logarithm of ln 3 with base 3, which is essential for transforming the expression into a more manageable form.

Change of Base Formula

The change of base formula allows the conversion of logarithms from one base to another, expressed as log_b(a) = log_k(a) / log_k(b) for any positive k. This is particularly useful when dealing with logarithms of different bases, such as converting log_3(ln 3) into a more familiar base, facilitating easier calculations and evaluations in the problem.