Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 2^x=64

Verified step by step guidance

1. Identify the bases on both sides of the equation: the left side is already base 2, and the right side is 64.
2. Express 64 as a power of 2. Since 64 is a power of 2, write 64 as \$2^{n}\( where \)n$ is an integer.
3. Rewrite the equation using the same base: \$2^{x} = 2^{n}$.
4. Since the bases are the same and the equation holds true, set the exponents equal to each other: \(x = n\).
5. Solve for \(x\) by determining the value of \(n\) from the expression of 64 as a power of 2.

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 the exponents directly.

Expressing Numbers as Powers of the Same Base

To solve exponential equations, rewrite each number as a power of a common base. For example, 64 can be expressed as 2^6, allowing the equation 2^x = 2^6 to be solved by equating exponents.

Equating Exponents

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