{"id":3163,"date":"2025-08-15T22:23:15","date_gmt":"2025-08-15T22:23:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3163"},"modified":"2025-09-09T20:57:14","modified_gmt":"2025-09-09T20:57:14","slug":"logarithmic-functions-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/logarithmic-functions-apply-it\/","title":{"raw":"Logarithmic Functions: Apply It","rendered":"Logarithmic Functions: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Convert between logarithmic to exponential form.<\/li>\r\n \t<li>Evaluate logarithms.<\/li>\r\n \t<li>Use common and natural logarithms<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Water Quality Assessment for Safe Swimming<\/h2>\r\n<p class=\"whitespace-normal break-words\">Maria is an environmental scientist working for the county health department. She tests water quality at public beaches and swimming areas to ensure they're safe for recreation. One of her key measurements is pH level, which indicates how acidic or basic the water is.<\/p>\r\n\r\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">The pH scale ranges from 0 (very acidic) to 14 (very basic), with 7 being neutral. Safe swimming water should have a pH between 7.2 and 7.8.\r\n\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">Maria needs to convert between pH readings and actual hydrogen ion concentrations to properly assess water safety and communicate results to the public.<\/p>\r\n\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">The relationship between pH and hydrogen ion concentration follows a logarithmic pattern: [latex]\\text{pH} = -\\log[H^+][\/latex]. Converting between logarithmic and exponential forms allows us to move between the convenient pH scale and the actual chemical concentrations that determine water safety.\r\n\r\n<\/section><section class=\"textbox recall\" aria-label=\"Recall\">\r\n<p class=\"whitespace-normal break-words\">To convert from logarithmic form to exponential form:<\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the logarithmic equation: [latex]\\text{pH} = -\\log[H^+][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Isolate the logarithm: [latex]-\\text{pH} = \\log[H^+][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Convert to exponential form: [latex][H^+] = 10^{-\\text{pH}}[\/latex]<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">Maria tests water at Sunset Beach and finds a pH of 8.2. What is the hydrogen ion concentration?<\/p>\r\n<p class=\"whitespace-normal break-words\">\r\n[reveal-answer q=\"194279\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"194279\"]Starting with the pH reading: [latex]\\text{pH} = 8.2[\/latex] Using the exponential form: [latex][H^+] = 10^{-\\text{pH}}[\/latex] [latex]\\begin{aligned} [H^+] &amp;= 10^{-8.2} \\ [H^+] &amp;= 6.31 \\times 10^{-9} \\text{ moles per liter} \\end{aligned}[\/latex] This very small concentration indicates the water is basic (pH &gt; 7) and safe for swimming.[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311271 [\/ohm_question]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Maria receives lab results showing a water sample has a hydrogen ion concentration of [latex][H^+] = 1.0 \\times 10^{-6}[\/latex] moles per liter. What is the pH?\r\n\r\n[reveal-answer q=\"683327\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"683327\"]Starting with the concentration: [latex][H^+] = 1.0 \\times 10^{-6}[\/latex] Using the logarithmic form: [latex]\\text{pH} = -\\log[H^+][\/latex] [latex]\\begin{aligned} \\text{pH} &amp;= -\\log(1.0 \\times 10^{-6}) \\ \\text{pH} &amp;= -\\log(10^{-6}) \\ \\text{pH} &amp;= -(-6) \\ \\text{pH} &amp;= 6 \\end{aligned}[\/latex] This pH of 6 indicates slightly acidic water, which may require treatment before swimming is safe.[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox recall\" aria-label=\"Recall\">When evaluating [latex]\\log(10^n) = n[\/latex], remember that the logarithm asks \"to what power must 10 be raised to get this number?\"\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311274[\/ohm_question]\r\n\r\n<\/section><\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Convert between logarithmic to exponential form.<\/li>\n<li>Evaluate logarithms.<\/li>\n<li>Use common and natural logarithms<\/li>\n<\/ul>\n<\/section>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Water Quality Assessment for Safe Swimming<\/h2>\n<p class=\"whitespace-normal break-words\">Maria is an environmental scientist working for the county health department. She tests water quality at public beaches and swimming areas to ensure they&#8217;re safe for recreation. One of her key measurements is pH level, which indicates how acidic or basic the water is.<\/p>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">The pH scale ranges from 0 (very acidic) to 14 (very basic), with 7 being neutral. Safe swimming water should have a pH between 7.2 and 7.8.<\/p>\n<\/section>\n<p class=\"whitespace-normal break-words\">Maria needs to convert between pH readings and actual hydrogen ion concentrations to properly assess water safety and communicate results to the public.<\/p>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">The relationship between pH and hydrogen ion concentration follows a logarithmic pattern: [latex]\\text{pH} = -\\log[H^+][\/latex]. Converting between logarithmic and exponential forms allows us to move between the convenient pH scale and the actual chemical concentrations that determine water safety.<\/p>\n<\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">\n<p class=\"whitespace-normal break-words\">To convert from logarithmic form to exponential form:<\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Identify the logarithmic equation: [latex]\\text{pH} = -\\log[H^+][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Isolate the logarithm: [latex]-\\text{pH} = \\log[H^+][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Convert to exponential form: [latex][H^+] = 10^{-\\text{pH}}[\/latex]<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">Maria tests water at Sunset Beach and finds a pH of 8.2. What is the hydrogen ion concentration?<\/p>\n<p class=\"whitespace-normal break-words\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q194279\">Show Solution<\/button><\/p>\n<div id=\"q194279\" class=\"hidden-answer\" style=\"display: none\">Starting with the pH reading: [latex]\\text{pH} = 8.2[\/latex] Using the exponential form: [latex][H^+] = 10^{-\\text{pH}}[\/latex] [latex]\\begin{aligned} [H^+] &= 10^{-8.2} \\ [H^+] &= 6.31 \\times 10^{-9} \\text{ moles per liter} \\end{aligned}[\/latex] This very small concentration indicates the water is basic (pH &gt; 7) and safe for swimming.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311271\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311271&theme=lumen&iframe_resize_id=ohm311271&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Maria receives lab results showing a water sample has a hydrogen ion concentration of [latex][H^+] = 1.0 \\times 10^{-6}[\/latex] moles per liter. What is the pH?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q683327\">Show Solution<\/button><\/p>\n<div id=\"q683327\" class=\"hidden-answer\" style=\"display: none\">Starting with the concentration: [latex][H^+] = 1.0 \\times 10^{-6}[\/latex] Using the logarithmic form: [latex]\\text{pH} = -\\log[H^+][\/latex] [latex]\\begin{aligned} \\text{pH} &= -\\log(1.0 \\times 10^{-6}) \\ \\text{pH} &= -\\log(10^{-6}) \\ \\text{pH} &= -(-6) \\ \\text{pH} &= 6 \\end{aligned}[\/latex] This pH of 6 indicates slightly acidic water, which may require treatment before swimming is safe.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">When evaluating [latex]\\log(10^n) = n[\/latex], remember that the logarithm asks &#8220;to what power must 10 be raised to get this number?&#8221;<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311274\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311274&theme=lumen&iframe_resize_id=ohm311274&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<\/div>\n<\/div>\n","protected":false},"author":67,"menu_order":22,"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\/3163"}],"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":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3163\/revisions"}],"predecessor-version":[{"id":3829,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3163\/revisions\/3829"}],"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\/3163\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3163"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3163"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3163"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3163"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}