Convert the following expressions to the indicated base.2x2ˣ using base e

Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Briggs 3rd Edition

Ch. 1 - Functions

문제 1.71

1장, 문제 1.71

Convert the following expressions to the indicated base.

$2 to the x-th power$ using base e

검증된 단계별 안내

Recognize that the expression $2^x$ needs to be converted to a base $e$ expression.

Recall the change of base formula: $a^x = e^{x \ln a}$.

Apply the change of base formula to $2^x$: $2^x = e^{x \ln 2}$.

Understand that $\ln 2$ is the natural logarithm of 2, which is a constant.

Conclude that the expression $2^x$ in terms of base $e$ is $e^{x \ln 2}$.

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

Exponential functions are mathematical expressions in the form of f(x) = a^x, where 'a' is a positive constant and 'x' is a variable. These functions exhibit rapid growth or decay and are characterized by their constant ratio of change. Understanding exponential functions is crucial for converting between different bases, as they form the foundation for many mathematical models in calculus.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base 'e', where 'e' is approximately equal to 2.71828. It is the inverse function of the exponential function with base 'e'. The natural logarithm is essential for converting expressions from one base to another, particularly when dealing with exponential growth or decay in calculus.

Change of Base Formula

The change of base formula allows for the conversion of logarithms from one base to another. It states that log_b(a) = log_k(a) / log_k(b) for any positive 'k' not equal to 1. This formula is particularly useful when converting exponential expressions, such as 2^x, to a different base like 'e', facilitating easier calculations and comparisons in calculus.