Logarithm Function

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We will now define the logarithm as the inverse function of the exponential function. As we have seen, the exponential function is bijective as a map from \(\mathbb{R}\) to \((0,\infty)\). This justifies the following definition.

In many books, the above defined function is also denoted by logarithmus naturalis \(\ln\). Before we collect some properties, a general result about continuity of inverse functions is presented.

Proof: First we show strict monotonic increase. Let \(y_1,y_2\in J\) with \(y_1< y_2\). Then \[f(f^{-1}(y_1))=y_1 < y_2 = f(f^{-1}(y_2))\] shows that, since \(f\) is strictly monotonically increasing, \(f^{-1}(y_1)\geq f^{-1}(y_2)\) cannot hold, so that \(f^{-1}(y_1) < f^{-1}(y_2)\). But this means that \(f^{-1}\) is strictly monotonically increasing.

Now we show continuity. Let \(\varepsilon>0\) and \(y_0\in J=f(I)\). Set \(x_0:=f^{-1}(y_0)\in I\). Since \(I\) is an open interval, there is an \(\varepsilon'>0\) with \(\varepsilon'<\varepsilon\) such that \([x_0-\varepsilon',x_0+\varepsilon']\subset I\). Since \(f\) is strictly monotonically increasing, \[\delta:=\min\{f(x_0+\varepsilon')-y_0,y_0-f(x_0-\varepsilon')\}>0.\] Then for \(y\in J\) with \(|y-y_0|<\delta\) holds \(f(x_0-\varepsilon')< y< f(x_0+\varepsilon')\) and the intermediate value theorem yields \[x:=f^{-1}(y) \in [x_0-\varepsilon',x_0+\varepsilon'] \subset ~ (x_0-\varepsilon,x_0+\varepsilon) ~ = ~ (f^{-1}(y_0)-\varepsilon,f^{-1}(y_0)+\varepsilon)\ ,\] i.e. \(|f^{-1}(y)-f^{-1}(y_0)|<\varepsilon\). By the \(\varepsilon\)-\(\delta\) criterion this means that \(f^{-1}\) is continuous in \(y_0\). \(\Box\)

We want to remark that Theorem 2 also holds for intervals \(I\) and \(J\) which are not open. In this case the proof of continuity of \(f^{-1}\) in \(y_0\in J\) has to be slightly adapted for boundary points \(x_0:=f^{-1}(y_0)\). This was dropped for simplicity reasons.