Problem 2 - College Algebra Essentials

Question

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$3^{x}=81$$

Step-by-Step Solution

Answer

The solution for the exponential equation \(3^{x} = 81\) is \(x = 4\).

Step 1: Express Both Sides with Same Base

The first step is to write both sides of the equation \(3^{x} = 81\) with the same base. As both 3 and 81 are powers of 3, rewrite 81 as \(3^{4}\). So the equation can be rewritten as \(3^{x} = 3^{4}\).

Step 2: Equating the Exponents

Once both sides of the equation have the same base, the exponents can be equated to each other. This results into a new equation which is \(x = 4\).

Step 3: Solving for x

Since \(x = 4\) is the solution of the equation, there is no further computation necessary.