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Exponential Approximations

Exponential Function

For all $x$,

The expansion as a sum implies that

You can see this in the figure. Around $x = 0$, the blue graph of $e^x$ and the red graph of $1+x$ are almost indistinguishable.

Log Function

The figure also shows that

This approximation is easier to understand if you write $x$ as 1 plus a small increment $w$. The approximation becomes

Now you can see the connection between this approximation and the one for the exponential function above. This one follows by taking logs on both sides of the approximation to $e^x$ for small $x$. It doesn’t matter whether you refer to the small number as $x$ or $w$. But it does matter that it is small.