Solve each equation. x = log3 1/81

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 20

5장, 문제 20

Solve each equation. x = log₃ 1/81

검증된 단계별 안내

Recognize that the equation is given in logarithmic form: \(x = \log_{3} \frac{1}{81}\). This means \(x\) is the exponent to which the base 3 must be raised to get \(\frac{1}{81}\).

Rewrite the equation in its equivalent exponential form: \(3^{x} = \frac{1}{81}\).

Express \(\frac{1}{81}\) as a power of 3. Since \(81 = 3^{4}\), then \(\frac{1}{81} = 3^{-4}\).

Set the exponents equal to each other because the bases are the same: \(3^{x} = 3^{-4}\) implies \(x = -4\).

Conclude that the solution to the equation is \(x = -4\).

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.

Logarithms and Their Definition

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log₃(1/81) asks for the power to which 3 must be raised to get 1/81. Understanding this definition is essential to solving logarithmic equations.

Properties of Exponents

Since logarithms are inverses of exponents, knowing how to manipulate exponents is crucial. Recognizing that 81 is 3 raised to the 4th power (3⁴ = 81) helps rewrite 1/81 as 3⁻⁴, simplifying the logarithmic expression.

Solving Logarithmic Equations

To solve equations like x = log₃(1/81), rewrite the logarithmic form into exponential form: 3ˣ = 1/81. Then, express 1/81 as a power of 3 and solve for x by equating exponents, which leads to the solution.