1.3: 1.3 The Logarithm Function
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 \[\begin{align} \ln(xy) &= \ln(x) + \ln(y) \\ \ln(x/y) &= \ln(x) - \ln(y).\end{align}\]