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 in base 2, and the right side, 32, can be expressed as a power of 2.
2. Rewrite 32 as a power of 2. Since 32 is equal to 2^5, rewrite the equation as 2^{2x-1} = 2^5.
3. Now that both sides of the equation have the same base, you can equate the exponents: 2x - 1 = 5.
4. Solve the equation 2x - 1 = 5 for x by first adding 1 to both sides to get 2x = 6.
5. Divide both sides by 2 to isolate x, resulting in x = 3.

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where variables appear in the exponent. 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.

Base and Exponent

In an exponential expression, the base is the number that is raised to a power, while the exponent indicates how many times the base is multiplied by itself. Understanding the relationship between bases and exponents is crucial for manipulating exponential equations, as it allows for the conversion of different forms of expressions into a common base.

Equating Exponents

When both sides of an exponential equation are expressed with the same base, the next step is to set the exponents equal to each other. This principle stems from the fact that if a^m = a^n (where a is the base), then m must equal n. This allows for straightforward algebraic solutions to find the value of the variable.