If ( log _{7}^{2}, 7^{log }left(e^{x}-5right), 7^{log _{7}left(2^{2}-frac{7}{2}right)} ) are in geometric progression then the sum of the possible values
Question
If ( log _{7}^{2}, 7^{log }left(e^{x}-5right), 7^{log _{7}left(2^{2}-frac{7}{2}right)} ) are in geometric progression then the sum of the possible values
Solution
1. Break Down the Problem
We need to analyze the given three terms:
We will determine the condition under which these terms are in geometric progression (GP). For three terms , they are in GP if .
2. Relevant Concepts
To prove the terms are in GP:
- Set
- Set
- Set (because )
3. Analysis and Detail
First, we express the condition for GP: This gives us: Now we will further analyze :
Knowing that , we can rewrite . Thus, we have: Next, we evaluate the term: Now, substituting this into the equation:
4. Verify and Summarize
Now we need to resolve: Once we perform this, we can derive and find values of satisfying the equation.
Final Answer
The computations and analysis indicate that upon solving these equations we can find possible values for . Evaluating these will yield a sum of valid values. Therefore:
The final result from solving will lead towards possible values which can be summarized as .
If you specifically evaluate solving and its two potential solutions, the final derived value series are total .
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