In Exercises 93–102, solve each equation. 3^x+2 ⋅ 3^x=81

Verified step by step guidance

1. Recognize that the equation is \(3^{x+2} \cdot 3^x = 81\). Use the property of exponents that states \(a^m \cdot a^n = a^{m+n}\) to combine the terms on the left side.
2. Rewrite the left side as \(3^{(x+2) + x} = 3^{2x + 2}\). So the equation becomes \(3^{2x + 2} = 81\).
3. Express 81 as a power of 3. Since \(81 = 3^4\), rewrite the equation as \(3^{2x + 2} = 3^4\).
4. Since the bases are the same and the equation \(3^{2x + 2} = 3^4\) holds, set the exponents equal to each other: \(2x + 2 = 4\).
5. Solve the linear equation \(2x + 2 = 4\) for \(x\) by isolating \(x\). Subtract 2 from both sides and then divide by 2.

Key Concepts

Exponential Functions

Exponential functions involve variables in the exponent, such as 3^x. Understanding their properties, like how to manipulate and simplify expressions with the same base, is essential for solving equations involving exponential terms.

Combining Like Terms with Exponents

When terms have the same base and variable exponent, they can often be combined by factoring. For example, 3^x + 2·3^x can be rewritten as 3·3^x, simplifying the equation and making it easier to solve.

Solving Exponential Equations by Equating Bases

To solve equations like 3·3^x = 81, express all terms with the same base. Since 81 = 3^4, set the exponents equal after simplifying, allowing you to solve for x using algebraic methods.