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 ln 3))

검증된 단계별 안내

Recognize that the function is given as \(f(x) = 3^x\) and you need to evaluate \(f(\log_3(2 \ln 3))\).

Recall the definition of the function evaluation: \(f(\log_3(2 \ln 3)) = 3^{\log_3(2 \ln 3)}\).

Use the property of exponents and logarithms that states \(a^{\log_a(b)} = b\) for any positive \(a \neq 1\) and \(b > 0\).

Apply this property to simplify \(3^{\log_3(2 \ln 3)}\) directly to \(2 \ln 3\).

Thus, the expression \(f(\log_3(2 \ln 3))\) simplifies to \(2 \ln 3\) without further calculation.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 a^(m+n) = a^m * a^n. Understanding how to manipulate and evaluate these functions is essential for solving problems involving exponents.

Logarithmic Functions

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

Properties of Logarithms and Exponents

The properties of logarithms and exponents, such as the inverse relationship a^(log_a(x)) = x and the ability to rewrite expressions using these properties, are crucial for simplifying complex expressions. For example, evaluating f(log_3(2 ln 3)) involves applying these properties to simplify the composition of functions.

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