Problem 118 - Introductory and Intermediate Algebra for College Students 4th

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 have the same base, rewrite both sides so that they are expressed as powers of this base, set the exponents equal to each other, and then solve the resulting equation.

Step 1: Express Both Sides as Powers of the Same Base

To begin, it is important to express both sides of the given equation as powers of the same base. Identify the common base that both terms of the equation can be raised to in order to express them equivalently. If the base terms of the equation are not the same, seek a way to rewrite them in terms of a common base.

Step 2: Setting the Exponents Equal to Each Other

Once both sides of the equation are expressed as powers of the same base, it can be concluded that the exponents must be equal to each other for the entire equation to hold true. This stems from the fundamental rule of exponents stating that if a^m = a^n, then m must equal n.

Step 3: Solve the Resulting Equation

With the exponents now set equal to each other, what remains is a simpler equation to solve. Standard methods of solving equations, like isolating variables and simplifying, can now be used. Finding the value of the variable in this step gives the answer of the original equation.