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 (2 ln 3))

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. Understanding their properties, such as the behavior of the function as 'x' approaches positive or negative infinity, is crucial for evaluating expressions involving exponents.

Logarithmic Functions

Logarithmic functions are the inverses 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. Key properties include the product, quotient, and power rules, which simplify the evaluation of logarithmic expressions and are essential for manipulating and solving equations involving logarithms.

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, as it enables easier computation and comparison. Understanding this formula is vital for evaluating logarithmic expressions in various contexts, including the given problem.