Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 8^(x+3)=16^(x−1)

Verified step by step guidance

1. Identify the bases on both sides of the equation: \(8^{(x+3)} = 16^{(x-1)}\). Notice that both 8 and 16 can be expressed as powers of 2.
2. Rewrite each base as a power of 2: \(8 = 2^3\) and \(16 = 2^4\). Substitute these into the equation to get \((2^3)^{(x+3)} = (2^4)^{(x-1)}\).
3. Apply the power of a power property: \((a^m)^n = a^{m \cdot n}\). This gives \(2^{3(x+3)} = 2^{4(x-1)}\).
4. Since the bases are the same (both are base 2), set the exponents equal to each other: \(3(x+3) = 4(x-1)\).
5. Solve the resulting linear equation for \(x\) by expanding both sides and isolating \(x\).

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 each number as a power of a common base. For example, 8 can be written as 2³ and 16 as 2⁴, enabling the comparison of exponents when bases match.

Equating Exponents

Once both sides of an equation have the same base, the exponents can be set equal to each other because if a^m = a^n, then m = n. This step transforms the problem into a simpler algebraic equation to solve for the variable.