Problem 121 - College Algebra Essentials

Question

Explain how to solve an exponential equation when both sides can be written as a power of the same base.

Step-by-Step Solution

Answer

To solve an exponential equation where both sides can be written as a power of the same base, identify a common base for both sides, then express the equation in this common base, which allows to equate the exponents resulting in a simpler equation. Solve this equation to find the solution to the original exponential equation.

Step 1: Identifying a Common Base

Start by looking at both sides of the given equation and try to identify a common base that can be used in expressing each side. The common base can be any number as long as both sides of the equation can be expressed in terms of this base.

Step 2: Expressing Both Sides with the Same Base

Once the common base is identified, go ahead and rewrite both sides of the equation using this common base. This will result in an equation of the form \(a^{x} = a^{y}\).

Step 3: Equating the Exponents

Since in an equation \(a^{x} = a^{y}\), the only way for this to be true is if \(x = y\), we can equate the two exponents. This will result in an equation without exponents.

Step 4: Solving the Resulting Equation

Solve the resulting equation obtained from the previous step. The solved value corresponds to the unknown in the original equation.