# Solved – Why do we use the natural exponential in logistic regression

I would like to intuitively understand the benefit of using the natural exponential in the sigmoid function used in logistic regression.

Why should it have to be \$e^x\$ instead of, for example \$2^x\$?

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Because base \$e\$ is convenient, and it doesn't matter if you can freely scale your coefficient estimate.

Would using a functional form of \$frac{a^mathbf{xcdot b}}{1 + a^mathbf{xcdot b} }\$ change your explanatory power? No.

### Explanation:

I gave basically the same answer here for the softmax function.

Observe that \$ e^ { mathbf{x} cdot mathbf{b} left( ln a right) } = a^ {mathbf{x} cdot mathbf{b}}\$. Hence:

\$\$ frac{a^mathbf{xcdot b}}{1 + a^mathbf{xcdot b} } = frac{e^mathbf{xcdot tilde{b}}}{1 + e^mathbf{xcdot tilde{b}} } \$\$

Where \$tilde{mathbf{b}} = left( ln a right) mathbf{b} \$. So using a different base than \$e\$ in the sigmoid function is the same as scaling your \$mathbf{b}\$ vector.

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