Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 2^2x-1=32

Verified step by step guidance

1. Recognize that the equation is \$2^{2x-1} = 32$. The goal is to express both sides as powers of the same base.
2. Recall that 32 can be written as a power of 2 because \$32 = 2^5$.
3. Rewrite the equation using this expression: \$2^{2x-1} = 2^5$.
4. Since the bases are the same and the equation holds true, set the exponents equal to each other: \$2x - 1 = 5$.
5. Solve the resulting linear equation for \(x\) by isolating \(x\).

Key Concepts

Exponential Equations

An exponential equation is one in which variables appear as exponents. Solving such equations often involves rewriting both sides to have the same base, allowing the exponents to be set equal to each other. This method simplifies the problem to solving a linear equation in the exponent.

Expressing Numbers as Powers of the Same Base

To solve exponential equations, it is crucial to rewrite each side as a power of the same base. For example, 32 can be expressed as 2^5 since 2 multiplied by itself 5 times equals 32. This step enables direct comparison of exponents.

Equating Exponents

Once both sides of an equation have the same base, the exponents can be set equal because if a^m = a^n, then m = n. This principle allows the conversion of an exponential equation into a simpler algebraic equation to solve for the variable.