Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 5^x=125

Verified step by step guidance

1. Recognize that the equation is an exponential equation of the form $5^x=\; 125$.
2. Express 125 as a power of 5. Since 125 = 5^3, rewrite the equation as $5^x=5^3$.
3. Since the bases on both sides of the equation are the same (base 5), set the exponents equal to each other: $x=3$.
4. Solve the resulting equation for $x$, which is straightforward since it is already isolated.
5. State the solution for $x$ based on the equality of the exponents.

Key Concepts

Exponential Equations

An exponential equation is an equation where variables appear as exponents. Solving such 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, 125 can be expressed as 5^3, allowing the equation 5^x = 5^3 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, such as x = 3.