Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 32^x=8

Verified step by step guidance

1. Identify the bases on both sides of the equation: the left side is ${32}^x$ and the right side is $8$.
2. Express both 32 and 8 as powers of the same base. Since both are powers of 2, rewrite them as $2^5$ for 32 and $2^3$ for 8.
3. Rewrite the equation using these powers: ${2^5}^x=2^3$.
4. Simplify the left side by multiplying the exponents: $2^{5\cdot x}=2^3$.
5. Since the bases are the same, set the exponents equal to each other: $5x\; =\; 3$, then solve for $x$.

Key Concepts

Exponential Equations

An exponential equation is an equation where variables appear as exponents. Solving such equations often involves rewriting expressions to have the same base, allowing the exponents to be set equal to each other for simplification.

Expressing Numbers as Powers of the Same Base

To solve exponential equations, rewrite each side as a power of the same base. For example, 32 and 8 can both be expressed as powers of 2 (32 = 2^5, 8 = 2^3), which helps in comparing and equating the exponents.

Equating Exponents

Once both sides of an exponential equation have the same base, the equation reduces to setting the exponents equal. This step transforms the problem into a simpler algebraic equation that can be solved for the variable.