What are these numbers trying to do?

Algebra Level 3

Challenge: Try to solve this without using logarithm!

If \(a, b, c, d\) and \(e\) are real values that satisfy the system of equations:

\(29^{a} = 31^{b} = 37^{d} = 41^{d} = 1363783^{23}\)

Find the value of \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d}\)

There is exactly one place in solving this problem when you really need a calculator, but not a scientific one. Round your answer to 5 decimal places.


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