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 (ln 3))

Verified step by step guidance

1. Identify the given function and the expression to evaluate: ƒ(x) = 3^x and we need to find ƒ(\(\log\)_3(\(\ln\) 3)).
2. Recall that ƒ(x) = 3^x means that when you input a value into ƒ, you raise 3 to the power of that value. So, ƒ(\(\log\)_3(\(\ln\) 3)) = 3^{\(\log\)_3(\(\ln\) 3)}.
3. Use the property of logarithms and exponents that states: for any positive a (a \(\neq\) 1), \ a^{\(\log\)_a(b)} = b. This means the base and the log base cancel out, leaving just the argument of the logarithm.
4. Apply this property to simplify 3^{\(\log\)_3(\(\ln\) 3)} to just \(\ln\) 3, because the base 3 and the log base 3 cancel out.
5. Conclude that ƒ(\(\log\)_3(\(\ln\) 3)) simplifies to \(\ln\) 3, which is the natural logarithm of 3.

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 properties such as f(0) = 1 and continuous increase or decrease depending on the base.

Logarithmic Functions and Their Properties

A logarithmic function is the inverse of an exponential function, defined as log_a(x), which answers the question: to what power must a be raised to get x? Key properties include log_a(a^x) = x and a^{log_a(x)} = x, which help simplify expressions involving logs and exponents.

Composition of Functions and Inverse Relationships

Composing functions involves applying one function to the result of another, such as f(g(x)). When dealing with exponential and logarithmic functions, their inverse relationship means that applying one after the other can simplify expressions, e.g., a^{log_a(x)} = x, which is essential for evaluating the given expression.