{"id":2182,"date":"2023-03-03T19:50:03","date_gmt":"2023-03-03T18:50:03","guid":{"rendered":"https:\/\/mxth.dk\/?p=2182"},"modified":"2026-04-09T22:25:30","modified_gmt":"2026-04-09T20:25:30","slug":"one-log-to-rule-them-all","status":"publish","type":"post","link":"https:\/\/mxth.dk\/?p=2182","title":{"rendered":"One log to Rule Them All"},"content":{"rendered":"\n<p class=\" eplus-wrapper\">Vi har nu v\u00e6ret igennem en masse forskellige logaritmefunktioner og vi vil her se p\u00e5 sammenh\u00e6ngen mellem de forskellige logaritmer.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Der findes en logaritme til hvert grundtal i en eksponentialfunktion, hvilket vil sige der er mange. Vi har nu med tilstedev\u00e6relsen af computeren of CAS-v\u00e6rkt\u00f8jer ikke noget problem i at skulle bruge logaritmer med forskellige basetal, men det ville v\u00e6re dejligt, hvis vi kunne n\u00f8jes med en enkelt eller to af dem.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">I gamle dage inden regnemaskibernes indtog var det n\u00e6sten en n\u00f8dvendighed fordi vi ellers skulle have en k\u00e6mpe tabel for hver at de mange logaritmer. Hvis man kunne n\u00f8jes med \u00e9n logaritme s\u00e5 var der kun \u00e9n tabel.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Men hvis vi skal kunne have kun \u00e9n logaritme at arbejde med s\u00e5 bliver vi n\u00f8d til at kunne omskrive \u00e9n logaritme til en anden.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Lad os som eksempel s\u00e5 p\u00e5 to logaritmer nemlig ti-talslogaritmen og den naturlige logaritme. Vi kunne have valgt alle mulige andre men vi ser som eksempel p\u00e5 disse to og vil senere g\u00f8res vores observationer mere generelt.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">P\u00e5 billedet herunder ses de to logaritmefunktioner indtegnet i et koordinatsystem.<\/p>\n\n\n<figure class=\" wp-block-image aligncenter size-large is-resized eplus-wrapper eplus-styles-uid-280c62\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/mxth.dk\/wp-content\/uploads\/2023\/03\/LogLink_ManimCE_v0.17.2.png?resize=600%2C338&#038;ssl=1\" alt=\"\" class=\"wp-image-2183\" width=\"600\" height=\"338\" srcset=\"https:\/\/i0.wp.com\/mxth.dk\/wp-content\/uploads\/2023\/03\/LogLink_ManimCE_v0.17.2.png?resize=1024%2C576&amp;ssl=1 1024w, https:\/\/i0.wp.com\/mxth.dk\/wp-content\/uploads\/2023\/03\/LogLink_ManimCE_v0.17.2.png?resize=300%2C169&amp;ssl=1 300w, https:\/\/i0.wp.com\/mxth.dk\/wp-content\/uploads\/2023\/03\/LogLink_ManimCE_v0.17.2.png?resize=768%2C432&amp;ssl=1 768w, https:\/\/i0.wp.com\/mxth.dk\/wp-content\/uploads\/2023\/03\/LogLink_ManimCE_v0.17.2.png?resize=1536%2C864&amp;ssl=1 1536w, https:\/\/i0.wp.com\/mxth.dk\/wp-content\/uploads\/2023\/03\/LogLink_ManimCE_v0.17.2.png?resize=2048%2C1152&amp;ssl=1 2048w, https:\/\/i0.wp.com\/mxth.dk\/wp-content\/uploads\/2023\/03\/LogLink_ManimCE_v0.17.2.png?w=3000&amp;ssl=1 3000w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/figure>\n\n\n<p class=\" eplus-wrapper\">Vi kan se at for x = 1 s\u00e5 sk\u00e6rer de hinanden. Ti-talslogaritmen er st\u00f8rre end den naturlige logaritme n\u00e5r x er mindre end 1 og den er mindre end den naturlige logaritme n\u00e5r x er st\u00f8rre end 1. Vi kunne derfor f\u00e5 tanken at hvis vi ganger vores ti-talslogaritme med et tal k s\u00e5 vil den blive til en naturlig logaritme.<\/p>\n\n\n<figure class=\" wp-block-image aligncenter size-full is-resized eplus-wrapper eplus-styles-uid-280c62\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/mxth.dk\/wp-content\/uploads\/2023\/03\/ezgif.com-video-to-gif7.gif?resize=600%2C338&#038;ssl=1\" alt=\"\" class=\"wp-image-2245\" width=\"600\" height=\"338\"><\/figure>\n\n\n<p class=\" eplus-wrapper\">P\u00e5 GIF\u2019en herunder ses hvorledes vores ti-talslogaritme \u00e6ndre sig n\u00e5r vi ganger den med et k som ligger mellem 1 og 3. Vi kan se at det p\u00e5 et tidspunkt er identisk med den naturlige logaritme. Lad os pr\u00f8ve at stoppe n\u00e5r de to logaritmefunktioner er ens.<\/p>\n\n\n<figure class=\" wp-block-image aligncenter size-full eplus-wrapper eplus-styles-uid-280c62\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"600\" height=\"338\" src=\"https:\/\/i0.wp.com\/mxth.dk\/wp-content\/uploads\/2023\/03\/ezgif.com-video-to-gif6.gif?resize=600%2C338&#038;ssl=1\" alt=\"\" class=\"wp-image-2246\"><\/figure>\n\n\n<p class=\" eplus-wrapper\">Vi har at hvis vi ganger vores ti-talslogaritme med 2,303 s\u00e5 for vi \u00e5benbart den naturlige logaritme. Sp\u00f8rgsm\u00e5let er bare hvor kommer de 2,303 fra? Uden vi har nogen anelse som hvor de kommer fra s\u00e5 lad os starte med et bud. <\/p>\n\n\n\n<p class=\" eplus-wrapper\"><code><div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-6\" class=\"h5p-iframe\" data-content-id=\"6\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"log_base_change_log10_to_ln\"><\/iframe><\/div><\/code><\/p>\n\n\n\n<p class=\" eplus-wrapper\"><\/p>\n\n\n\n<p class=\" eplus-wrapper\">Det vil sige, at hvis vi skal finde den konstant der omdanner en ti-talslogaritme til den naturlige logaritme skal vi alts\u00e5 tage den logaritme vi vil omdanne <em>til<\/em> (i dette tilf\u00e6lde den naturlige logaritme) til base for den vi vil omdanne <em>fra<\/em> (i dette tilf\u00e6lde ti-talslogaritmen). Og vi har derfor at at de 2,303 svare til ln(10) &#8211; alts\u00e5 hvor mange gange skal e ganges med sig selv for at f\u00e5 10.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Hvis vi skal omskrive ti-talslogaritmen til den naturlige logaritme har vi derfor at<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(\\ln(x)=\\log_{10}(x)\\cdot\\ln(10)\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">eller hvis vi samler ens logaritmer p\u00e5 samme side<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(\\log_{10}(x)=\\frac{\\ln(x)}{\\ln(10)}\\)<\/p>\n\n\n<div class=\"wp-block-group has-background-color has-primary-background-color has-text-color has-background eplus-wrapper is-layout-constrained wp-block-group-is-layout-constrained eplus-styles-uid-2f00ed\"><p class=\" eplus-wrapper eplus-styles-uid-1eb5dc\">S\u00e6tning<\/p>\n\n<hr class=\"is-style-wide wp-block-separator has-alpha-channel-opacity eplus-wrapper eplus-styles-uid-9ce347\">\n\n<p class=\" eplus-wrapper eplus-styles-uid-2946e1\">Vi har derfor generelt at omskrivningen mellem en logaritme og en anden f\u00f8lger f\u00f8lgende lighed<\/p>\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(\\log_a(x)=\\frac{\\log_b(x)}{\\log_b(a)}\\)<\/p>\n<\/div>\n\n\n<p class=\" eplus-wrapper\">Du kan herunder pr\u00f8ve et par omskrivninger.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-8\" class=\"h5p-iframe\" data-content-id=\"8\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Opgaver_til_change_log_base\"><\/iframe><\/div><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Vi har nu v\u00e6ret igennem en masse forskellige logaritmefunktioner og vi vil her se p\u00e5 sammenh\u00e6ngen mellem de forskellige logaritmer. Der findes en logaritme til hvert grundtal i en eksponentialfunktion, hvilket vil sige der er mange. Vi har nu med tilstedev\u00e6relsen af computeren of CAS-v\u00e6rkt\u00f8jer ikke noget problem i at skulle bruge logaritmer med forskellige [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","editor_plus_copied_stylings":"{}","_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[12,36,3],"tags":[],"class_list":["post-2182","post","type-post","status-publish","format-standard","hentry","category-funktioner","category-logaritmefunktioner","category-matematik"],"featured_image_src":null,"author_info":{"display_name":"Henriksen","author_link":"https:\/\/mxth.dk\/?author=1"},"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/2182","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2182"}],"version-history":[{"count":23,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/2182\/revisions"}],"predecessor-version":[{"id":2249,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/2182\/revisions\/2249"}],"wp:attachment":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2182"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2182"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2182"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}