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Physics LibreTexts

1.3: 1.3 The Logarithm Function

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Since the exponential is a one-to-one function, its inverse is a well-defined function. We call this the natural logarithm: ln(x)ysuch thatexp(y)=x.

For brevity, we will henceforth use “logarithm” to refer to the natural logarithm, unless otherwise stated (the “non-natural” logarithms are not our concern in this course). The domain of the logarithm is yR+, and its range is R. Its graph is shown below:

clipboard_e7d023381aed6495c4ed48278be7fef1c.png
Figure 1.3.1

Observe that the graph increases extremely slowly with x, precisely the opposite of the exponential’s behavior.

Using Eq. (1.2.3), we can prove that the logarithm satisfies the product and quotient rules ln(xy)=ln(x)+ln(y)ln(x/y)=ln(x)ln(y).


This page titled 1.3: 1.3 The Logarithm Function is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to the style and standards of the LibreTexts platform.

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