Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-26T09:11:38.807Z Has data issue: false hasContentIssue false
  • English
  • Français

The real solutions of x = ax

Published online by Cambridge University Press:  22 June 2022

A. F. Beardon
A. F. Beardon*
Affiliation:
Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

We denote the real logarithm of a positive number a by ln a, so that ax = exp (x ln a), and we shall discuss what is known about the real solutions x of the equation (1)

$$x = {a^x},\;\,a > 0.$$
First, as exp t > 0 for all real t, each real solution x of (1) is positive.

Type
Articles
Information
The Mathematical Gazette , Volume 106 , Issue 566 , July 2022 , pp. 206 - 211
Check for updates
Copyright
© The Authors, 2022 Published by Cambridge University Press on behalf of The Mathematical Association

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Stroud, L. J., On x = ax, Math. Gaz. 53 (October 1969) pp. 289-293.Google Scholar
Beardon, A. F. and Gordon, R. A., The convexity of the function y = E (x) defined by xy = yx, Math. Gaz. 104 (March 2020) pp. 36-43.Google Scholar
Rippon, P. J., Infinite exponentials, Math. Gaz. 67 (October 1983) 189-196.Google Scholar
Euler, L., De formulis exponentialibus replicatis, Opera Omnia, Series Prima XV (1927) pp. 268-297; Acta Acad. Petropolitanae 1 (1777) 38-60.Google Scholar
Knoebel, R. A., Exponentials reiterated, Amer. Math. Monthly 88 (1981) pp. 235–252.Google Scholar
Baker, I. N. and Rippon, P. J., A note on complex iteration, Amer. Math. Monthly 92 (1985) pp. 501–504.Google Scholar
Galidakis, I. N., On an application of Lambert’s W function to infinite exponentials, Complex Variables 49 (2004) pp. 759–780.Google Scholar
Beardon, A. F., The principal branch of the Lambert W Function, Comp. Methods and Fn. Thy (2021), pp. 307–316.Google Scholar
Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J. and Knuth, D. E., On the Lambert W function, Adv. Comput. Math. 5 (1966) 329–359.Google Scholar