Solve each equation. Give solutions in exact form. log2 (log2 x) = 1

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Lial 13th Edition

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

문제 81

5장, 문제 81

Solve each equation. Give solutions in exact form. log₂ (log₂ x) = 1

검증된 단계별 안내

Start by understanding the equation: \(\log_2 (\log_2 x) = 1\). This means the logarithm base 2 of another logarithm base 2 of \(x\) equals 1.

Rewrite the equation by setting an inner variable: let \(y = \log_2 x\). Then the equation becomes \(\log_2 y = 1\).

Solve for \(y\) by converting the logarithmic equation to its exponential form: \(y = 2^1\).

Substitute back \(y = \log_2 x\) to get \(\log_2 x = 2\).

Solve for \(x\) by rewriting the logarithmic equation in exponential form: \(x = 2^2\).

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.

Properties of Logarithms

Understanding the properties of logarithms, such as the definition log_b(a) = c means b^c = a, is essential. This allows you to rewrite logarithmic equations into exponential form to solve for the variable.

Domain Restrictions of Logarithmic Functions

Logarithmic functions are only defined for positive arguments. When solving nested logarithmic equations like log_2(log_2 x), you must ensure both the inner and outer logarithm arguments are positive to find valid solutions.

Solving Nested Logarithmic Equations

Solving equations with nested logarithms involves working from the outer logarithm inward. First, isolate the inner logarithm, then solve the resulting simpler logarithmic or exponential equation step-by-step.