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

Verified step by step guidance

1. Recognize that the equation is \(3^{1-x} = \frac{1}{27}\). The goal is to express both sides with the same base.
2. Recall that \(27\) can be written as a power of \(3\) because \$27 = 3^3$. Therefore, rewrite the right side as \(\frac{1}{3^3}\).
3. Use the property of exponents that \(\frac{1}{a^n} = a^{-n}\) to rewrite \(\frac{1}{3^3}\) as \$3^{-3}$.
4. 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$.
5. Solve the equation \$1 - x = -3\( for \)x\( by isolating \)x$ on one side.

Key Concepts

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.