Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 3^1-x=1/27

검증된 단계별 안내

Recognize that the equation is \(3^{1-x} = \frac{1}{27}\). The goal is to express both sides with the same base.

Recall that \(27\) can be written as a power of \(3\) because \(27 = 3^3\). Therefore, rewrite the right side as \(\frac{1}{3^3}\).

Use the property of exponents that \(\frac{1}{a^n} = a^{-n}\) to rewrite \(\frac{1}{3^3}\) as \(3^{-3}\).

Now the equation is \(3^{1-x} = 3^{-3}\). Since the bases are the same and the function is one-to-one, set the exponents equal: \(1 - x = -3\).

Solve the equation \(1 - x = -3\) for \(x\) by isolating \(x\) on one side.

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 Equations

An exponential equation is one in which variables appear as exponents. Solving these equations often involves rewriting expressions so that both sides have the same base, allowing the exponents to be set equal to each other.

Expressing Numbers as Powers of the Same Base

To solve exponential equations, it is essential to rewrite each side as a power of the same base. For example, 1/27 can be expressed as 3 to the power of -3, since 27 = 3^3 and the reciprocal is 3^-3.

Equating Exponents

Once both sides of an equation have the same base, the exponents can be set equal to each other because if a^m = a^n and a ≠ 0 or 1, then m = n. This step simplifies the problem to solving a linear equation in the exponent.

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 31-x=1/27