Problem 121 - College Algebra

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

For exponential equations where both sides have a common base, simplify the equation by setting the exponents equal to each other and then solve for the variable. For instance, in the equation \(2^{3x} = 2^{7}\) after equating the exponents we obtain \(3x = 7\), and therefore \(x = 7/3\).

Step 1: Identification

Understand the question and identify the exponential equations where both sides can be written as a power of the same base. Let's consider an example for better understanding: \(2^{3x} = 2^{7}\). Here, \(2^{3x}\) and \(2^{7}\) are exponential notations where both sides can be written as a power of same base, which is 2.

Step 2: Reduce to Common Base

When the bases are equal, we can equate exponents. Here, because both terms have a base of 2, the equation is simplified and we get: \(3x = 7\).

Step 3: Solve for the variable

We have a linear equation at this stage, which we can solve using elementary algebra. Divide both sides of the equation by 3 to isolate x: \(x = 7/3\).