Solving Exponential Equations: Step-by-Step Guide

Solving Exponential Equations

In this lecture, we address solving equations involving exponents, specifically the example:
Let ( x^5 = 3^{10} ). The goal is to find the value of ( x ).

Analyzing the Equation

When examining the equation, we notice the left-hand side has a power of 5 (i.e., ( x^5 )) while the right-hand side has a power of 10 (i.e., ( 3^{10} )). The immediate objective is to manipulate the right-hand side so that both sides of the equation contain the same exponent.

Manipulating the Right-Hand Side

To match the powers, we need to express ( 3^{10} ) in terms of power 5. This can be achieved by rewriting 10 in a way that relates to 5:

• Recognize that ( 10 = 2 \times 5 ).
• Therefore, we can rewrite ( 3^{10} ) as ( 3^{2 \cdot 5} ).
• In exponential notation, this is equivalent to:
  [ 3^{10} = (3^2)^5 ]

Reformulating the Equation

Now substituting back into the original equation gives us:
[ x^5 = (3^2)^5 ]
This simplification allows us to analyze the equation more easily.

Evaluating the Exponents

Next, we simplify the right-hand side:

• Calculating ( 3^2 ) results in:
  [ 3^2 = 3 \times 3 = 9 ]
  Thus, we can rewrite the equation as:
  [ x^5 = 9^5 ]

Canceling the Exponents

At this stage, since both sides of the equation now have the same exponent of 5, we can cancel these exponents:

• This cancellation gives:
  [ x = 9 ]
  Therefore, the solution to the equation ( x^5 = 3^{10} ) is ( x = 9 ).

Conclusion

Thus, the final answer to the given equation is:
[ x = 9 ]

This step-by-step method demonstrates the importance of manipulating exponents and recognizing equivalent forms to solve exponential equations effectively.
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