Problem 121
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
Verified Answer
To solve an exponential equation with both sides expressed as powers of the same base, the solution involves identifying the common base, equating the exponents, and then solving the resulting equation.
1Step 1: Identify the common base
Firstly, it is essential to identify a common base for both sides of the equation. In many cases, this involves factoring or simplifying the equation.
2Step 2: Equate the exponents
Once both sides of the equation have been expressed as a power of the same base, the next step is to set the exponents of the two sides equal to each other. This is justified by the fact that if \(a^b = a^c\) with \(a>0\) and \(a≠1\), then \(b=c\). This results in a simpler equation to solve.
3Step 3: Solve the equation
The equation from Step 2 can be solved using standard algebraic techniques. Depending on the complexity of the exponent equations, you may need to use techniques such as the distributive property or factoring.
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