Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 9^x=27

Verified step by step guidance

1. Identify the bases on both sides of the equation: the left side is 9^x and the right side is 27.
2. Express both 9 and 27 as powers of the same base. Since both 9 and 27 are powers of 3, rewrite them as 9 = 3^2 and 27 = 3^3.
3. Rewrite the equation using these expressions: (3^2)^x = 3^3.
4. Apply the power of a power rule by multiplying the exponents on the left side: 3^(2x) = 3^3.
5. Since the bases are the same, set the exponents equal to each other: 2x = 3, and then solve for x.

Key Concepts

Exponential Equations

An exponential equation is an equation where variables appear as exponents. Solving such equations often involves rewriting both sides with the same base to compare the exponents directly.

Expressing Numbers as Powers of a Common Base

To solve exponential equations, rewrite each number as a power of the same base. For example, 9 can be written as 3^2 and 27 as 3^3, allowing the equation to be expressed with a common base.

Equating Exponents

Once both sides of an exponential equation have the same base, their exponents can be set equal to each other. This reduces the problem to solving a simpler algebraic equation involving the exponents.