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As per my understanding of logistic regression, a log of odds of the desired value of “y” should be in linear relation with the log (x). Does that mean that independent variables should have exponential distribution, which is again a non-normal distribution?

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No, the logistic model is actually simpler than you are describing. The log of odds of the predicted value of “y” should be in linear relation simply with "x". And "x" might follow any of a variety of distributions.

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