{"id":1148,"date":"2025-07-23T16:27:15","date_gmt":"2025-07-23T16:27:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1148"},"modified":"2025-09-15T16:13:16","modified_gmt":"2025-09-15T16:13:16","slug":"logarithmic-properties-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/logarithmic-properties-apply-it\/","title":{"raw":"Logarithmic Properties: Apply It","rendered":"Logarithmic Properties: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Expand logarithmic expressions.<\/li>\r\n \t<li>Condense logarithmic expressions.<\/li>\r\n \t<li>Use the change-of-base formula for logarithms.<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox connectIt\" aria-label=\"Connect It\">\r\n\r\n[caption id=\"\" align=\"alignright\" width=\"244\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010829\/CNX_Precalc_Figure_04_05_001F2.jpg\" alt=\"Testing of the pH of hydrochloric acid.\" width=\"244\" height=\"382\" \/> The pH of hydrochloric acid is tested with litmus paper. (credit: David Berardan)[\/caption]\r\n<p id=\"fs-id1165137759741\">In chemistry, <strong>pH<\/strong> is used as a measure of the acidity or alkalinity of a substance. The pH scale runs from 0 to 14. Substances with a pH less than 7 are considered acidic, and substances with a pH greater than 7 are said to be alkaline. Our bodies, for instance, must maintain a pH close to 7.35 in order for enzymes to work properly. To get a feel for what is acidic and what is alkaline, consider the following pH levels of some common substances:<\/p>\r\n\r\n<ul id=\"fs-id1165135253210\">\r\n \t<li>Battery acid: 0.8<\/li>\r\n \t<li>Stomach acid: 2.7<\/li>\r\n \t<li>Orange juice: 3.3<\/li>\r\n \t<li>Pure water: 7 (at 25\u00b0 C)<\/li>\r\n \t<li>Human blood: 7.35<\/li>\r\n \t<li>Fresh coconut: 7.8<\/li>\r\n \t<li>Sodium hydroxide (lye): 14<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137540406\">To determine whether a solution is acidic or alkaline, we find its pH, which is a measure of the number of active positive hydrogen ions in the solution. The pH is defined by the following formula, where <em>a<\/em>\u00a0is the concentration of hydrogen ion in the solution<\/p>\r\n\r\n<div id=\"eip-396\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}\\text{pH}&amp;=-\\mathrm{log}\\left(\\left[{H}^{+}\\right]\\right) \\\\ &amp;=\\mathrm{log}\\left(\\frac{1}{\\left[{H}^{+}\\right]}\\right) \\end{align}[\/latex]<\/div>\r\n<p id=\"fs-id1165137472164\">The equivalence of [latex]-\\mathrm{log}\\left(\\left[{H}^{+}\\right]\\right)[\/latex] and [latex]\\mathrm{log}\\left(\\frac{1}{\\left[{H}^{+}\\right]}\\right)[\/latex] is one of the logarithm properties we will examine in this section.<\/p>\r\n\r\n<\/section>\r\n<p data-start=\"332\" data-end=\"398\">We know that<br data-start=\"344\" data-end=\"347\" \/>[latex]\\text{pH} = -\\log\\left([H^+]\\right)[\/latex].<\/p>\r\n<p data-start=\"400\" data-end=\"481\">Let\u2019s explore how the logarithm properties help us interpret and compare acidity.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"526\" data-end=\"606\">Suppose a solution has [latex][H^+] = 3.2 \\times 10^{-4}[\/latex]. Let's use properties of logarithms to simplify its\u00a0pH.<\/p>\r\n<p data-start=\"526\" data-end=\"606\">[latex]\\begin{align}\r\n\\text{pH} &amp;= -\\log\\big(3.2\\times 10^{-4}\\big)\r\n&amp;&amp; \\text{substitute into pH formula} \\\\\r\n&amp;= -\\Big(\\log(3.2) + \\log(10^{-4})\\Big)\r\n&amp;&amp; \\text{product rule: }\\log(ab)=\\log a + \\log b \\\\\r\n&amp;= -\\Big(\\log(3.2) - 4,\\log(10)\\Big)\r\n&amp;&amp; \\text{power rule: }\\log(10^{-4})=-4\\log(10) \\\\\r\n&amp;= -\\log(3.2) + 4\\log(10)\r\n&amp;&amp; \\text{distribute the negative sign} \\\\\r\n&amp;= -\\log(3.2) + 4\r\n&amp;&amp; \\text{since }\\log(10)=1\u00a0 \\\\\r\n\\end{align}[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312046[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Expand logarithmic expressions.<\/li>\n<li>Condense logarithmic expressions.<\/li>\n<li>Use the change-of-base formula for logarithms.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">\n<figure style=\"width: 244px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010829\/CNX_Precalc_Figure_04_05_001F2.jpg\" alt=\"Testing of the pH of hydrochloric acid.\" width=\"244\" height=\"382\" \/><figcaption class=\"wp-caption-text\">The pH of hydrochloric acid is tested with litmus paper. (credit: David Berardan)<\/figcaption><\/figure>\n<p id=\"fs-id1165137759741\">In chemistry, <strong>pH<\/strong> is used as a measure of the acidity or alkalinity of a substance. The pH scale runs from 0 to 14. Substances with a pH less than 7 are considered acidic, and substances with a pH greater than 7 are said to be alkaline. Our bodies, for instance, must maintain a pH close to 7.35 in order for enzymes to work properly. To get a feel for what is acidic and what is alkaline, consider the following pH levels of some common substances:<\/p>\n<ul id=\"fs-id1165135253210\">\n<li>Battery acid: 0.8<\/li>\n<li>Stomach acid: 2.7<\/li>\n<li>Orange juice: 3.3<\/li>\n<li>Pure water: 7 (at 25\u00b0 C)<\/li>\n<li>Human blood: 7.35<\/li>\n<li>Fresh coconut: 7.8<\/li>\n<li>Sodium hydroxide (lye): 14<\/li>\n<\/ul>\n<p id=\"fs-id1165137540406\">To determine whether a solution is acidic or alkaline, we find its pH, which is a measure of the number of active positive hydrogen ions in the solution. The pH is defined by the following formula, where <em>a<\/em>\u00a0is the concentration of hydrogen ion in the solution<\/p>\n<div id=\"eip-396\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}\\text{pH}&=-\\mathrm{log}\\left(\\left[{H}^{+}\\right]\\right) \\\\ &=\\mathrm{log}\\left(\\frac{1}{\\left[{H}^{+}\\right]}\\right) \\end{align}[\/latex]<\/div>\n<p id=\"fs-id1165137472164\">The equivalence of [latex]-\\mathrm{log}\\left(\\left[{H}^{+}\\right]\\right)[\/latex] and [latex]\\mathrm{log}\\left(\\frac{1}{\\left[{H}^{+}\\right]}\\right)[\/latex] is one of the logarithm properties we will examine in this section.<\/p>\n<\/section>\n<p data-start=\"332\" data-end=\"398\">We know that<br data-start=\"344\" data-end=\"347\" \/>[latex]\\text{pH} = -\\log\\left([H^+]\\right)[\/latex].<\/p>\n<p data-start=\"400\" data-end=\"481\">Let\u2019s explore how the logarithm properties help us interpret and compare acidity.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"526\" data-end=\"606\">Suppose a solution has [latex][H^+] = 3.2 \\times 10^{-4}[\/latex]. Let&#8217;s use properties of logarithms to simplify its\u00a0pH.<\/p>\n<p data-start=\"526\" data-end=\"606\">[latex]\\begin{align}  \\text{pH} &= -\\log\\big(3.2\\times 10^{-4}\\big)  && \\text{substitute into pH formula} \\\\  &= -\\Big(\\log(3.2) + \\log(10^{-4})\\Big)  && \\text{product rule: }\\log(ab)=\\log a + \\log b \\\\  &= -\\Big(\\log(3.2) - 4,\\log(10)\\Big)  && \\text{power rule: }\\log(10^{-4})=-4\\log(10) \\\\  &= -\\log(3.2) + 4\\log(10)  && \\text{distribute the negative sign} \\\\  &= -\\log(3.2) + 4  && \\text{since }\\log(10)=1\u00a0 \\\\  \\end{align}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312046\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312046&theme=lumen&iframe_resize_id=ohm312046&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":13,"menu_order":11,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":510,"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\/1148"}],"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\/13"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1148\/revisions"}],"predecessor-version":[{"id":3993,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1148\/revisions\/3993"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/510"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1148\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1148"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1148"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1148"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1148"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}