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 of the exponential expressions on both sides of the equation: \(9^x = 27\).
2. Express both 9 and 27 as powers of the same base. Since both are powers of 3, write \(9 = 3^2\) and \(27 = 3^3\).
3. Rewrite the equation using these expressions: \((3^2)^x = 3^3\).
4. Apply the power of a power property by multiplying the exponents: \(3^{2x} = 3^3\).
5. Since the bases are the same and the expressions are equal, set the exponents equal to each other: \(2x = 3\), then solve for \(x\).

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 helpful to rewrite each number as a power of a common base. For example, 9 can be written as 3² and 27 as 3³, enabling comparison of exponents when bases match.

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, then m = n. This principle simplifies solving for the variable in the exponent.