{"id":11,"date":"2004-02-23T20:10:53","date_gmt":"2004-02-23T20:10:53","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=11"},"modified":"2004-02-23T20:10:53","modified_gmt":"2004-02-23T20:10:53","slug":"powers-and-logs","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=11","title":{"rendered":"Powers and Logs"},"content":{"rendered":"<p>Prove that <img src='\/maths\/latexrender\/pictures\/463e729014deb4ca69afa6b27a34f67d.gif' title='2^{\\ln3}=3^{\\ln2}' alt='2^{\\ln3}=3^{\\ln2}' align=absmiddle><br \/>\nThe proof should show you how this generalises. If you have studied group theory you can extend this even further by showing that <img src='\/maths\/latexrender\/pictures\/c27b0972287a6a36afec97ccd4c665d7.gif' title='\\left(\\mathbb{R}^{+}-\\left\\{1\\right\\},\\circ\\right)' alt='\\left(\\mathbb{R}^{+}-\\left\\{1\\right\\},\\circ\\right)' align=absmiddle>, where <img src='\/maths\/latexrender\/pictures\/1b3c1a40f9cb094d47e8c6f9b0df773f.gif' title='\\circ' alt='\\circ' align=absmiddle> is defined by <img src='\/maths\/latexrender\/pictures\/9c3396885b2755c2f6371e888ab5c11e.gif' title='\\begin{displaymath} x\\circ y=x^{\\ln y}\\end{displaymath}' alt='\\begin{displaymath} x\\circ y=x^{\\ln y}\\end{displaymath}' align=absmiddle>, is an abelian group.<br \/>\nWhy can&#8217;t 1 be an element of this group?<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Prove that The proof should show you how this generalises. If you have studied group theory you can extend this even further by showing that , where is defined by , is an abelian group. Why can&#8217;t 1 be an element of this group?<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-11","post","type-post","status-publish","format-standard","hentry","category-articles"],"_links":{"self":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/11","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11"}],"version-history":[{"count":0,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/11\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}