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. Identify the bases on both sides of the equation: the left side is already a power of 2, and the right side, 32, can be expressed as a power of 2 because 32 = 2^5.
2. Rewrite the equation with both sides having the same base: $2^{\mathrm{2x\; -\; 1}}=2^5$.
3. Since the bases are the same and the equation is $2^{2x\; -\; 1}=2^5$, set the exponents equal to each other: $2x\; -\; 1\; =\; 5$.
4. Solve the linear equation for x: add 1 to both sides to get $2x\; =\; 6$, then divide both sides by 2 to isolate x.
5. Write the solution for x as $x\; =\; 3$ (do not calculate the final value here, just the steps).

Key Concepts

Exponential Equations

An exponential equation is one in which variables appear as exponents. Solving these equations often involves rewriting both sides with the same base to compare exponents directly. This approach simplifies the equation into a linear form that can be solved using algebraic methods.

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, 32 can be expressed as 2^5 since 2 raised to the 5th power equals 32. This step allows the exponents to be set equal to each other, facilitating the solution.

Equating Exponents

Once both sides of an exponential equation have the same base, their exponents can be set equal because if a^m = a^n, then m = n. This principle reduces the problem to solving a simpler algebraic equation involving the exponents, such as 2x - 1 = 5.