\n 1. Logarithmic Graph Examples<\/b><\/a>\n
\n\t 2. Relationship with Exponential Graph<\/a><\/font>\n <\/div>\n
\n
The exponential function and the logarithmic function and their relationship were featured on the exponential function and logarithms<\/i><\/a> introduction page.<\/p>\n On this Page: 1. Logarithmic Graph Examples 2. Relationship with Exponential Graph The exponential function and the logarithmic function and their relationship were featured on the exponential function and logarithms introduction page. When dealing with real numbers. Logarithmic Function. , . Exponential function. , The functions… Read More »Logarithmic Function Graph Examples<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"neve_meta_sidebar":"","neve_meta_container":"","neve_meta_enable_content_width":"","neve_meta_content_width":0,"neve_meta_title_alignment":"","neve_meta_author_avatar":"","neve_post_elements_order":"","neve_meta_disable_header":"","neve_meta_disable_footer":"","neve_meta_disable_title":"","footnotes":""},"wf_page_folders":[18],"class_list":["post-1030","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.learnermath.com\/wp-json\/wp\/v2\/pages\/1030","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnermath.com\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.learnermath.com\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnermath.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnermath.com\/wp-json\/wp\/v2\/comments?post=1030"}],"version-history":[{"count":18,"href":"https:\/\/www.learnermath.com\/wp-json\/wp\/v2\/pages\/1030\/revisions"}],"predecessor-version":[{"id":3660,"href":"https:\/\/www.learnermath.com\/wp-json\/wp\/v2\/pages\/1030\/revisions\/3660"}],"wp:attachment":[{"href":"https:\/\/www.learnermath.com\/wp-json\/wp\/v2\/media?parent=1030"}],"wp:term":[{"taxonomy":"wf_page_folders","embeddable":true,"href":"https:\/\/www.learnermath.com\/wp-json\/wp\/v2\/wf_page_folders?post=1030"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nWhen dealing with real numbers.\n
\nLogarithmic Function. f(x) \\space {\\small{=}} \\space{\\tt{log}}_a (x)<\/span><\/font> , a > 0 \\space \\space , \\space \\space a \\space {\\cancel{=}} \\space 1 \\space \\space , \\space \\space x > 0<\/span>.\n
\nExponential function. f(x) = a^x<\/span><\/font>, a > 0<\/span>\n
\nThe functions are inverses of each other.\n
\n{\\tt{log}}_a (a^x) \\space {\\small{=}} \\space x<\/span><\/font> => a^{{\\tt{log}}_a (x)} \\space {\\small{=}} \\space x<\/span><\/font> \n
\nHere we will look at some logarithmic function graph examples. Observing the standard form of the graphs, and how they differ on a cartesian axis.\n
\n<\/a>\n
\n\n\n\n
\n\n\n\nLogarithmic Function Graph
Examples<\/span><\/h2>\n\n\n\n
\nWhen we have, f(x) \\space {\\small{=}} \\space{\\tt{log}}_a (x)<\/span><\/font> , a > 0 \\space \\space , \\space \\space a \\space {\\cancel{=}} \\space 1 \\space \\space , \\space \\space x > 0<\/span>.\n
\nThe shape of the graph will be influenced by a<\/i><\/font><\/font>.\n
\n\n\n\na > 1<\/span><\/h3>\n\n\n\nLet’s firstly look at the case of, a > 1<\/span>.\n
\nf(x) \\space {\\small{=}} \\space {\\tt{log}}_2 (x)<\/span><\/font>\n
\nWe can obtain some points on the graph.\n
\nf(\\frac{1}{4}) \\space {\\small{=}} \\space {\\tt{log}}_2 (\\frac{1}{4}) \\space {\\small{=}} \\space {\\text{-}}2 \\space \\space \\space \\space \\space \\space \\space \\space {\\footnotesize{( 2^{{\\text{-}}2} = \\frac{1}{4} )}}<\/span><\/font>\n
\nf(\\frac{1}{2}) \\space {\\small{=}} \\space {\\tt{log}}_2 (\\frac{1}{2}) \\space {\\small{=}} \\space {\\text{-}}1 \\space \\space \\space \\space \\space \\space \\space \\space {\\footnotesize{( 2^{{\\text{-}}1} = \\frac{1}{2} )}}<\/span><\/font>\n
\nf(1) \\space {\\small{=}} \\space {\\tt{log}}_2 (1) \\space {\\small{=}} \\space 0 \\space \\space \\space \\space \\space \\space \\space \\space \\space \\space {\\footnotesize{( 2^0 = 1 )}}<\/span><\/font>\n
\nf(2) \\space {\\small{=}} \\space {\\tt{log}}_2 (2) \\space {\\small{=}} \\space 1 \\space \\space \\space \\space \\space \\space \\space \\space \\space \\space {\\footnotesize{( 2^1 = 2 )}}<\/span><\/font>\n
\nf(4) \\space {\\small{=}} \\space {\\tt{log}}_2 (4) \\space {\\small{=}} \\space 2 \\space \\space \\space \\space \\space \\space \\space \\space \\space \\space {\\footnotesize{( 2^2 = 4 )}}<\/span><\/font>\n
\nSo we have found the points (\\frac{1}{4},{\\text{-}}2) , (\\frac{1}{2},{\\text{-}}1) , (1,0) , (2,1) , (4,2)<\/span><\/font>.\n
\nThis information can now help in drawing the graph for f(x) \\space {\\small{=}} \\space {\\tt{log}}_2 (x)<\/span><\/font>.\n
\n\n\n\n![\"First](\"http:\/\/www.learnermath.com\/wp-content\/uploads\/2023\/09\/logarithmic-function-graph-examples.png\")
<\/figure>\n\n\n\n
\nThis logarithm graph above is the standard shape for a function f(x) \\space {\\small{=}} \\space {\\tt{log}}_a (x)<\/span> where a > 1<\/span>.\n
\nWe can observe the following.\n
\n–<\/b><\/font> As x<\/span> gets smaller towards 0, the graph trends to negative infinity.\n
\n–<\/b><\/font> As x<\/span> gets larger, the graph trends to positive infinity.\n
\n–<\/b><\/font> The graph is increasing and doesn’t cross the y<\/span>-axis.\n
\n–<\/b><\/font> The graph will pass through (1,0)<\/font>, and for f(x) \\space {\\small{=}} \\space {\\tt{log}}_a (x)<\/span>, will pass through (a<\/i><\/font>,1)<\/font>.\n
\nHere is another diagram with 2 other log function graphs for a > 1<\/span>.\n
\nWhere we can see how the shape is affected by a varying value of a<\/i><\/font>.\n
\n\n\n\n![\"Two](\"http:\/\/www.learnermath.com\/wp-content\/uploads\/2023\/09\/a-greater-one-log-graphs.png\")
<\/figure>\n\n\n\n
\n\n\n\na < 0 < 1<\/span><\/h3>\n\n\n\nThe other case for logarithm graphs is, 0 \\lt a \\lt 1<\/span>.\n
\nWe can examine f(x) \\space {\\small{=}} \\space {\\tt{log}}_{\\frac{1}{2}} (x)<\/span><\/font>.\n
\nAgain obtaining some points on the graph.\n
\nf(\\frac{1}{4}) \\space {\\small{=}} \\space {\\tt{log}}_{\\frac{1}{2}} (\\frac{1}{4}) \\space {\\small{=}} \\space 2 \\space \\space \\space \\space \\space \\space \\space \\space \\space \\space {\\footnotesize{( \\space (\\frac{1}{2})^2 = \\frac{1}{4} \\space )}}<\/span><\/font>\n
\nf(\\frac{1}{2}) \\space {\\small{=}} \\space {\\tt{log}}_{\\frac{1}{2}} (\\frac{1}{2}) \\space {\\small{=}} \\space 1 \\space \\space \\space \\space \\space \\space \\space \\space \\space \\space {\\footnotesize{( \\space (\\frac{1}{2})^1 = \\frac{1}{2} \\space )}}<\/span><\/font>\n
\nf(1) \\space {\\small{=}} \\space {\\tt{log}}_{\\frac{1}{2}} (1) \\space {\\small{=}} \\space 0 \\space \\space \\space \\space \\space \\space \\space \\space \\space \\space \\space {\\footnotesize{( \\space (\\frac{1}{2})^0 = 1 \\space )}}<\/span><\/font>\n
\nf(2) \\space {\\small{=}} \\space {\\tt{log}}_{\\frac{1}{2}} (2) \\space {\\small{=}} \\space {\\text{-}}1 \\space \\space \\space \\space \\space \\space \\space \\space \\space \\space {\\footnotesize{( \\space (\\frac{1}{2})^{{\\text{-}}1} = 2 \\space )}}<\/span><\/font>\n
\nf(4) \\space {\\small{=}} \\space {\\tt{log}}_{\\frac{1}{2}} (4) \\space {\\small{=}} \\space {\\text{-}}2 \\space \\space \\space \\space \\space \\space \\space \\space \\space \\space {\\footnotesize{( \\space (\\frac{1}{2})^{{\\text{-}}2} = 4 \\space )}}<\/span><\/font>\n
\nWe have found the points (\\frac{1}{4},2) , (\\frac{1}{2},1) , (1,0) , (2,{\\text{-}}1) , (4,{\\text{-}}2)<\/span><\/font>.\n
\nThis information can now help in drawing the graph for f(x) \\space {\\small{=}} \\space {\\tt{log}}_{\\frac{1}{2}} (x)<\/span><\/font>.\n
\n\n\n\n![\"Logarithm](\"http:\/\/www.learnermath.com\/wp-content\/uploads\/2023\/09\/log-graph-a-less-one.png\")
<\/figure>\n\n\n\n
\nThis logarithm graph above is the standard shape for a function f(x) \\space {\\small{=}} \\space {\\tt{log}}_a (x)<\/span> where 0 \\lt a \\lt 1<\/span>.\n
\nThe following can be observed.\n
\n–<\/b><\/font> As x<\/span> gets smaller towards 0, the graph trends to positive infinity.\n
\n–<\/b><\/font> As x<\/span> gets larger, the graph trends to negative infinity.\n
\n–<\/b><\/font> The graph is decreasing and doesn’t cross the y<\/span>-axis.\n
\n–<\/b><\/font> The graph will pass through (1,0)<\/font>, and for f(x) \\space {\\small{=}} \\space {\\tt{log}}_a (x)<\/span>, will pass through (a<\/i><\/font>,1)<\/font>.\n
\nLike before, we can view another diagram with 2 other log function graphs for 0 \\lt a \\lt 1<\/span>.\n
\nObserving how the graph shape is affected by a varying value of a<\/i><\/font>.\n
\n\n\n\n![\"Two](\"http:\/\/www.learnermath.com\/wp-content\/uploads\/2023\/09\/couple-examples-a-under-one.png\")
<\/figure>\n\n\n\n
\n<\/a>\n
\n\n\n\n
\n\n\n\nRelationship with the Exponential Function Graph<\/span><\/h2>\n\n\n\n
\nA logarithmic function is the inverse of its corresponding exponential function.\n
\nAs such, the graph of the functions are reflected in the line y = x<\/span>.\n
\nThe coordinates get swapped around.\n
\n\n\n\n![\"Logarithmic](\"https:\/\/www.learnermath.com\/wp-content\/uploads\/2024\/04\/exponential-log-inverse-graph.png\")
<\/figure>\n\n\n\n
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