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 \$5^{x} = 125$. The goal is to express both sides as powers of the same base.
2. Recall that 125 can be written as a power of 5 because \$125 = 5^{3}$.
3. Rewrite the equation using this expression: \$5^{x} = 5^{3}$.
4. Since the bases are the same and the equation holds true, set the exponents equal to each other: \(x = 3\).
5. This gives the solution for \(x\) without needing to calculate any further.

Key Concepts

Exponential Equations

An exponential equation is one in which variables appear as exponents. Solving these equations often involves rewriting expressions so that both sides have the same base, allowing the exponents to be set equal to each other.

Expressing Numbers as Powers of the Same Base

To solve exponential equations, it is helpful to rewrite numbers as powers of a common base. For example, 125 can be expressed as 5³, which allows the equation 5^x = 125 to be rewritten as 5^x = 5^3.

Equating Exponents

Once both sides of an exponential equation have the same base, the exponents can be set equal to each other because if a^m = a^n, then m = n. This property simplifies solving for the variable in the exponent.