The logarithmic scale. Let us consider once more the series of functions \[x,\quad \sqrt{x},\quad \sqrt[3]{x},\ \dots,\quad \sqrt[n]{x},\ \dots,\] which possesses the property that, if \(f(x)\) and \(\phi(x)\) are any two of the functions contained in it, then \(f(x)\) and \(\phi(x)\) both tend to \(\infty\) as \(x \to \infty\), while \(f(x)/\phi(x)\) tends to \(0\) or to \(\infty\) according as \(f(x)\) occurs to the right or the left of \(\phi(x)\) in the series. We can now continue this series by the insertion of new terms to the right of all those already written down. We can begin with \(\log x\), which tends to infinity more slowly than any of the old terms. Then \(\sqrt{\log x}\) tends to \(\infty\) more slowly than \(\log x\), \(\sqrt[3]{\log x}\) than \(\sqrt{\log x}\), and so on. Thus we obtain a series \[x,\quad \sqrt{x},\quad \sqrt[3]{x},\ \dots,\quad \sqrt[n]{x},\ \dots\quad \log x,\quad \sqrt{\log x},\quad \sqrt[3]{\log x},\ \dots\quad \sqrt[n]{\log x},\ \dots\] formed of two simply infinite series arranged one after the other. But this is not all. Consider the function \(\log\log x\), the logarithm of \(\log x\). Since \((\log x)/x^{\alpha} \to 0\), for all positive values of \(\alpha\), it follows on putting \(x = \log y\) that \[(\log\log y)/(\log y)^{\alpha} = (\log x)/x^{\alpha} \to 0.\] Thus \(\log\log y\) tends to \(\infty\) with \(y\), but more slowly than any power of \(\log y\). Hence we may continue our series in the form \[\begin{gathered} x,\quad \sqrt{x},\quad \sqrt[3]{x},\ \dots\qquad \log x,\quad \sqrt{\log x},\quad \sqrt[3]{\log x},\ \dots\\ \log\log x,\quad \sqrt{\log\log x},\ \dots\quad \sqrt[n]{\log\log x},\ \dots;\end{gathered}\] and it will by now be obvious that by introducing the functions \(\log\log\log x\), \(\log\log\log\log x\), … we can prolong the series to any extent we like. By putting \(x = 1/y\) we obtain a similar scale of infinity for functions of \(y\) which tend to \(\infty\) as \(y\) tends to \(0\) by positive values.1

Example LXXXIV
1. Between any two terms \(f(x)\)\(F(x)\) of the series we can insert a new term \(\phi(x)\) such that \(\phi(x)\) tends to \(\infty\) more slowly than \(f(x)\) and more rapidly than \(F(x)\). [Thus between \(\sqrt{x}\) and \(\sqrt[3]{x}\) we could insert \(x^{5/12}\): between \(\sqrt{\log x}\) and \(\sqrt[3]{\log x}\) we could insert \((\log x)^{5/12}\). And, generally, \(\phi(x) = \sqrt{f(x) F(x)}\) satisfies the conditions stated.]

2. Find a function which tends to \(\infty\) more slowly than \(\sqrt{x}\), but more rapidly than \(x^{\alpha}\), where \(\alpha\) is any rational number less than \(1/2\). [\(\sqrt{x}/(\log x)\) is such a function; or \(\sqrt{x}/(\log x)^{\beta}\), where \(\beta\) is any positive rational number.]

3. Find a function which tends to \(\infty\) more slowly than \(\sqrt{x}\), but more rapidly than \(\sqrt{x}/(\log x)^{\alpha}\), where \(\alpha\) is any rational number. [The function \(\sqrt{x}/(\log\log x)\) is such a function. It will be gathered from these examples that incompleteness is an inherent characteristic of the logarithmic scale of infinity.]

4. How does the function \[f(x) = \{x^{\alpha} (\log x)^{\alpha’} (\log\log x)^{\alpha”}\}/ \{x^{\beta} (\log x)^{\beta’} (\log\log x)^{\beta”}\}\] behave as \(x\) tends to \(\infty\)? [If \(\alpha \neq \beta\) then the behaviour of \[f(x) = x^{\alpha-\beta} (\log x)^{\alpha’-\beta’} (\log\log x)^{\alpha”-\beta”}\] is dominated by that of \(x^{\alpha-\beta}\). If \(\alpha = \beta\) then the power of \(x\) disappears and the behaviour of \(f(x)\) is dominated by that of \((\log x)^{\alpha’-\beta’}\), unless \(\alpha’ = \beta’\), when it is dominated by that of \((\log\log x)^{\alpha”-\beta”}\). Thus \(f(x) \to \infty\) if \(\alpha > \beta\), or \(\alpha = \beta\), \(\alpha’ > \beta’\), or \(\alpha = \beta\), \(\alpha’ = \beta’\), \(\alpha” > \beta”\), and \(f(x) \to 0\) if \(\alpha < \beta\), or \(\alpha = \beta\), \(\alpha’ < \beta’\), or \(\alpha = \beta\), \(\alpha’ = \beta’\), \(\alpha” < \beta”\).]

5. Arrange the functions \(x/\sqrt{\log x}\), \(x\sqrt{\log x}/\log\log x\), \(x\log\log x/\sqrt{\log x}\), \((x\log\log\log x)/\sqrt{\log\log x}\) according to the rapidity with which they tend to infinity as \(x \to \infty\).

6. Arrange \[\log\log x/(x\log x),\quad (\log x)/x,\quad x\log\log x/\sqrt{x^{2} + 1},\quad \{\sqrt{x + 1}\}/x(\log x)^{2}\] according to the rapidity with which they tend to zero as \(x \to \infty\).

7. Arrange \[x\log\log(1/x),\quad \sqrt{x}/\{\log(1/x)\},\quad \sqrt{x\sin x\log(1/x)},\quad (1 – \cos x)\log(1/x)\] according to the rapidity with which they tend to zero as \(x \to +0\).

8. Show that \[D_{x}\log\log x = 1/(x\log x),\quad D_{x}\log\log\log x = 1/(x\log x\log\log x),\] and so on.

9. Show that \[D_{x}(\log x)^{\alpha} = \alpha/\{x(\log x)^{1-\alpha}\},\quad D_{x}(\log\log x)^{\alpha} = \alpha/\{x\log x(\log\log x)^{1-\alpha}\},\] and so on.

  1. For fuller information as to ‘scales of infinity’ see the author’s tract ‘Orders of Infinity’, Camb. Math. Tracts, No. 12.↩︎

$\leftarrow$ 199–201. The behaviour of $\log x$ as $x$ tends to infinity or to zero Main Page 203. The number \(e\) $\rightarrow$