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 and 16. Notice that both 8 and 16 can be expressed as powers of 2, since 8 = 2^3 and 16 = 2^4.
2. Rewrite each side of the equation using base 2: $8^{x+3}={16}^{x-1}$ becomes ${2^3}^{x+3}={2^4}^{x-1}$.
3. Use the power of a power property $(a^\{m\})^\{n\}\; =\; a^\{mn\}$ to simplify both sides: $2^\{3(x+3)\}\; =\; 2^\{4(x-1)\}$.
4. Since the bases are the same and the expressions are equal, set the exponents equal to each other: $3(x+3)\; =\; 4(x-1)$.
5. Solve the resulting linear equation for $x$: expand both sides, collect like terms, and isolate $x$.

Key Concepts

Exponential Equations

An exponential equation is one in which variables appear as exponents. Solving such equations often involves rewriting expressions to have the same base, allowing the exponents to be set equal. This approach simplifies the problem to solving a linear equation in the exponent.

Expressing Numbers as Powers of the Same Base

To solve exponential equations, it is crucial to rewrite each side as a power of the same base. For example, 8 can be written as 2^3 and 16 as 2^4. This common base allows the exponents to be compared directly, facilitating the solution process.

Equating Exponents

Once both sides of an equation have the same base, the exponents can be set equal because if a^m = a^n and a ≠ 0 or 1, then m = n. This principle reduces the problem to solving a simpler algebraic equation involving the exponents.