Properties of the logarithm

Problem

State for the function \(f\) with \(f(x)=\ln(2013-x)\) the maximal domain \(\mathbb{D}\), the behavior of \(f\) at the borders of \(\mathbb{D}\) as well as the intersections of the graph of \(f\) with the coordinate axes.

Solution

The domain of the natural logarithm \(\ln(x)\) is \((0,\infty)\). As a consequence, the domain of \(\ln(-x)\) is \((-\infty,0)\). Adding a number to the argument of the logarithm, the same needs to be done for the domain. Thus, the domain of \(f(x)=\ln(2013-x)\) is given by \(\mathbb{D}=(-\infty, 2013)\).

By means of Sage, we obtain an idea of the function graph.

At the borders of \(\mathbb{D}\) one finds the following behavior:

\[\lim\limits_{x\rightarrow -\infty}\ln(2013-x) = \lim\limits_{x\rightarrow -\infty}\ln(-x) = \lim\limits_{\tilde{x}\rightarrow +\infty}\ln(\tilde{x}) = +\infty\]

This result can be confirmed by means of Sage by inserting \(-\infty\) for \(x\).

For \(x\) going to \(2013\), the argument of the logarithm goes to \(0\). In this case, the logarithm goes to \(-\infty\).

\[\lim\limits_{x\rightarrow 2013^{-}}\ln(2013-x) = \lim\limits_{\tilde{x}\rightarrow 0^{+}}\ln(\tilde{x}) = -\infty\]

The intersection with the \(y\)-axis is obtained by setting \(x=0\):

\[f(0)=\ln(2013-0) = \ln(2013) \approx 7.61\]

This result is in agreement with the function graph shown above.

The logarithm intersects the \(x\)-axis if its argument equals \(1\):

\[2013-x \overset{!}{=}1 \rightarrow x = 2012\]

This result can be confirmed with Sage: