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1.3: 1.3 The Logarithm Function

Last updated

Apr 30, 2021
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- 1.2: 1.2 The Exponential Function
- 1.4: Non-Natural Powers

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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:

$\label{eq:1}\ln(x) \equiv y \;\; \mathrm{such}~\mathrm{that}\; \exp(y) = x.$ 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

$y \in \mathbb{R}^+$ , and its range is

$\mathbb{R}$ . Its graph is shown below:

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

$\begin{align} \ln(xy) &= \ln(x) + \ln(y) \\ \ln(x/y) &= \ln(x) - \ln(y).\end{align}$

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