{"id":3166,"date":"2025-08-15T22:23:50","date_gmt":"2025-08-15T22:23:50","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3166"},"modified":"2025-09-10T20:49:24","modified_gmt":"2025-09-10T20:49:24","slug":"graphs-of-logarithmic-functions-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-logarithmic-functions-apply-it\/","title":{"raw":"Graphs of Logarithmic Functions: Apply It","rendered":"Graphs of Logarithmic Functions: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Identify the domain of a logarithmic function.<\/li>\r\n \t<li>Graph logarithmic functions.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"text-2xl font-bold mt-1 text-text-100\">Logarithmic Functions in Sound Engineering<\/h2>\r\n<p data-start=\"326\" data-end=\"596\">Sound engineers use logarithmic functions to model sound intensity, frequency response, and acoustic properties. Graphing logarithmic transformations helps visualize sound distribution patterns and optimize audio equipment placement for the best audience experience.<\/p>\r\n\r\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">\r\n<h3>Why Logarithms are Used in Sound<\/h3>\r\n\r\n<hr \/>\r\n\r\nHuman hearing spans an enormous range\u2014from the faintest audible whisper ([latex]10^-12 W\/m^2[\/latex]) to the threshold of pain ([latex]1 W\/m^2[\/latex]). The decibal (dB) scale compresses this range using a base-10 logarithm:\r\n\r\n[latex]L=10log_10(\\frac{I}{I_0})[\/latex]\r\n\r\nwhere [latex]I[\/latex] is sound intensity and [latex]I_0=10^-12 W\/m^2[\/latex] is the reference intensity.\r\n\r\n<\/section><section class=\"textbox recall\" aria-label=\"Recall\">\r\n<p data-start=\"1121\" data-end=\"1141\">The parent function [latex]f(x) = \\log_{10}(x)[\/latex]<\/p>\r\n<p data-start=\"1121\" data-end=\"1141\">has the following characteristics:<\/p>\r\n\r\n<ul>\r\n \t<li data-start=\"1121\" data-end=\"1141\"><strong data-start=\"1209\" data-end=\"1220\">Domain:<\/strong>\u00a0<span style=\"color: #204a77;\">[latex](0,\\infty)[\/latex]<\/span><\/li>\r\n \t<li data-start=\"1121\" data-end=\"1141\"><strong data-start=\"1251\" data-end=\"1261\">Range:<\/strong> <span style=\"color: #204a77;\">[latex](-\\infty,\\infty)[\/latex]<\/span>]<\/li>\r\n \t<li data-start=\"1121\" data-end=\"1141\"><strong data-start=\"1298\" data-end=\"1321\">Vertical asymptote:<\/strong> [latex]x=0[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311311[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p data-start=\"2578\" data-end=\"2672\">[ohm_question hide_question_numbers=1]311312[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311331[\/ohm_question]<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"3671\" data-end=\"3703\">Compare these three functions:<\/p>\r\n\r\n<ul data-start=\"3705\" data-end=\"3850\">\r\n \t<li data-start=\"3705\" data-end=\"3752\">\r\n<p data-start=\"3707\" data-end=\"3752\">[latex]f_1(x) = 2\\log_{10}(x+1) - 2[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"3753\" data-end=\"3801\">\r\n<p data-start=\"3755\" data-end=\"3801\">[latex]f_2(x) = -2\\log_{10}(x+1) - 2[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"3802\" data-end=\"3850\">\r\n<p data-start=\"3804\" data-end=\"3850\">[latex]f_3(x) = 2\\log_{10}(-x-1) - 2[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<ol data-start=\"3868\" data-end=\"4131\">\r\n \t<li data-start=\"3868\" data-end=\"3933\">\r\n<p data-start=\"3871\" data-end=\"3933\">Identify the domain, asymptote, and vertical shift for each.<\/p>\r\n<\/li>\r\n \t<li data-start=\"3934\" data-end=\"3984\">\r\n<p data-start=\"3937\" data-end=\"3984\">Graph all three on the same coordinate plane.<\/p>\r\n<\/li>\r\n \t<li data-start=\"3985\" data-end=\"4084\">\r\n<p data-start=\"3988\" data-end=\"4084\">Which two curves could represent real-world EQ settings? Which one is only a math exploration?<\/p>\r\n<\/li>\r\n \t<li data-start=\"4085\" data-end=\"4131\">\r\n<p data-start=\"4088\" data-end=\"4131\">Explain your reasoning in terms of audio.<\/p>\r\n<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"613045\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"613045\"]\r\n\r\nDomains, asymptotes, vertical shifts:\r\n\r\n[latex]f_1[\/latex]:\r\n\r\nDomain [latex](-1,\\infty)[\/latex];\r\nasymptote [latex]x=-1[\/latex];\r\nvertical shift [latex]-2[\/latex].\r\n\r\n[latex]f_2[\/latex]:\r\n\r\nDomain [latex](-1,\\infty) [\/latex];\r\nasymptote [latex]x=-1[\/latex];\r\nvertical shift [latex]-2[\/latex];\r\nreflection across x-axis.\r\n\r\n[latex]f_3[\/latex]:\r\n\r\nDomain [latex](-\\infty,-1) [\/latex];\r\nasymptote [latex]x=-1[\/latex];\r\nvertical shift [latex]-2[\/latex];\r\nreflection across y-axis\r\n\r\nGraphing note: [latex]f_1[\/latex] and [latex]f_2[\/latex] share the same asymptote and horizontal shift; [latex]f_2[\/latex] is the vertical reflection of [latex]f_1[\/latex].\r\n\r\nReal-world use: [latex]f_1[\/latex] and [latex]f_2[\/latex] can represent EQ curves; [latex]f_3[\/latex] is only a math exploration because audio frequency\/intensity inputs require [latex]x&gt;-1[\/latex] (after the shift) to keep the log argument positive; [latex]f_3[\/latex] forces negative inputs, which don\u2019t correspond to physical frequencies\/intensities.[\/hidden-answer]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Identify the domain of a logarithmic function.<\/li>\n<li>Graph logarithmic functions.<\/li>\n<\/ul>\n<\/section>\n<h2 class=\"text-2xl font-bold mt-1 text-text-100\">Logarithmic Functions in Sound Engineering<\/h2>\n<p data-start=\"326\" data-end=\"596\">Sound engineers use logarithmic functions to model sound intensity, frequency response, and acoustic properties. Graphing logarithmic transformations helps visualize sound distribution patterns and optimize audio equipment placement for the best audience experience.<\/p>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">\n<h3>Why Logarithms are Used in Sound<\/h3>\n<hr \/>\n<p>Human hearing spans an enormous range\u2014from the faintest audible whisper ([latex]10^-12 W\/m^2[\/latex]) to the threshold of pain ([latex]1 W\/m^2[\/latex]). The decibal (dB) scale compresses this range using a base-10 logarithm:<\/p>\n<p>[latex]L=10log_10(\\frac{I}{I_0})[\/latex]<\/p>\n<p>where [latex]I[\/latex] is sound intensity and [latex]I_0=10^-12 W\/m^2[\/latex] is the reference intensity.<\/p>\n<\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">\n<p data-start=\"1121\" data-end=\"1141\">The parent function [latex]f(x) = \\log_{10}(x)[\/latex]<\/p>\n<p data-start=\"1121\" data-end=\"1141\">has the following characteristics:<\/p>\n<ul>\n<li data-start=\"1121\" data-end=\"1141\"><strong data-start=\"1209\" data-end=\"1220\">Domain:<\/strong>\u00a0<span style=\"color: #204a77;\">[latex](0,\\infty)[\/latex]<\/span><\/li>\n<li data-start=\"1121\" data-end=\"1141\"><strong data-start=\"1251\" data-end=\"1261\">Range:<\/strong> <span style=\"color: #204a77;\">[latex](-\\infty,\\infty)[\/latex]<\/span>]<\/li>\n<li data-start=\"1121\" data-end=\"1141\"><strong data-start=\"1298\" data-end=\"1321\">Vertical asymptote:<\/strong> [latex]x=0[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311311\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311311&theme=lumen&iframe_resize_id=ohm311311&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p data-start=\"2578\" data-end=\"2672\"><iframe loading=\"lazy\" id=\"ohm311312\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311312&theme=lumen&iframe_resize_id=ohm311312&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311331\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311331&theme=lumen&iframe_resize_id=ohm311331&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"3671\" data-end=\"3703\">Compare these three functions:<\/p>\n<ul data-start=\"3705\" data-end=\"3850\">\n<li data-start=\"3705\" data-end=\"3752\">\n<p data-start=\"3707\" data-end=\"3752\">[latex]f_1(x) = 2\\log_{10}(x+1) - 2[\/latex]<\/p>\n<\/li>\n<li data-start=\"3753\" data-end=\"3801\">\n<p data-start=\"3755\" data-end=\"3801\">[latex]f_2(x) = -2\\log_{10}(x+1) - 2[\/latex]<\/p>\n<\/li>\n<li data-start=\"3802\" data-end=\"3850\">\n<p data-start=\"3804\" data-end=\"3850\">[latex]f_3(x) = 2\\log_{10}(-x-1) - 2[\/latex]<\/p>\n<\/li>\n<\/ul>\n<ol data-start=\"3868\" data-end=\"4131\">\n<li data-start=\"3868\" data-end=\"3933\">\n<p data-start=\"3871\" data-end=\"3933\">Identify the domain, asymptote, and vertical shift for each.<\/p>\n<\/li>\n<li data-start=\"3934\" data-end=\"3984\">\n<p data-start=\"3937\" data-end=\"3984\">Graph all three on the same coordinate plane.<\/p>\n<\/li>\n<li data-start=\"3985\" data-end=\"4084\">\n<p data-start=\"3988\" data-end=\"4084\">Which two curves could represent real-world EQ settings? Which one is only a math exploration?<\/p>\n<\/li>\n<li data-start=\"4085\" data-end=\"4131\">\n<p data-start=\"4088\" data-end=\"4131\">Explain your reasoning in terms of audio.<\/p>\n<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q613045\">Show Solution<\/button><\/p>\n<div id=\"q613045\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domains, asymptotes, vertical shifts:<\/p>\n<p>[latex]f_1[\/latex]:<\/p>\n<p>Domain [latex](-1,\\infty)[\/latex];<br \/>\nasymptote [latex]x=-1[\/latex];<br \/>\nvertical shift [latex]-2[\/latex].<\/p>\n<p>[latex]f_2[\/latex]:<\/p>\n<p>Domain [latex](-1,\\infty)[\/latex];<br \/>\nasymptote [latex]x=-1[\/latex];<br \/>\nvertical shift [latex]-2[\/latex];<br \/>\nreflection across x-axis.<\/p>\n<p>[latex]f_3[\/latex]:<\/p>\n<p>Domain [latex](-\\infty,-1)[\/latex];<br \/>\nasymptote [latex]x=-1[\/latex];<br \/>\nvertical shift [latex]-2[\/latex];<br \/>\nreflection across y-axis<\/p>\n<p>Graphing note: [latex]f_1[\/latex] and [latex]f_2[\/latex] share the same asymptote and horizontal shift; [latex]f_2[\/latex] is the vertical reflection of [latex]f_1[\/latex].<\/p>\n<p>Real-world use: [latex]f_1[\/latex] and [latex]f_2[\/latex] can represent EQ curves; [latex]f_3[\/latex] is only a math exploration because audio frequency\/intensity inputs require [latex]x>-1[\/latex] (after the shift) to keep the log argument positive; [latex]f_3[\/latex] forces negative inputs, which don\u2019t correspond to physical frequencies\/intensities.<\/p><\/div>\n<\/div>\n<\/section>\n","protected":false},"author":67,"menu_order":28,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":105,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3166"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3166\/revisions"}],"predecessor-version":[{"id":3843,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3166\/revisions\/3843"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/105"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3166\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3166"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3166"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3166"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3166"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}