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 both sides of the equation can be expressed as powers of the same base. Notice that 27 is a power of 3, specifically, $27=3^3$.
2. Rewrite $\frac{1}{27}$ as $3^{-3}$ because $\frac{1}{a^b}=a^{-b}$.
3. Now the equation becomes $3^{1-x}=3^{-3}$.
4. Since the bases are the same, set the exponents equal to each other: $1-x=-3$.
5. Solve the equation $1-x=-3$ for $x$ by isolating $x$.

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where variables appear as exponents. To solve these equations, one common method is to express both sides with the same base, allowing for the exponents to be equated. This approach simplifies the problem and makes it easier to isolate the variable.

Properties of Exponents

The properties of exponents, such as the product of powers, power of a power, and the quotient of powers, are essential for manipulating exponential expressions. For instance, recognizing that 1/27 can be rewritten as 3^(-3) helps in equating the bases. Understanding these properties allows for effective simplification and solving of exponential equations.

Equating Exponents

Once both sides of an exponential equation are expressed with the same base, the next step is to equate the exponents. This principle stems from the fact that if a^m = a^n, then m must equal n, provided a is not zero. This step is crucial for isolating the variable and finding its value in the context of the equation.