{"id":11202,"date":"2015-07-14T18:50:13","date_gmt":"2015-07-14T18:50:13","guid":{"rendered":"https:\/\/courses.candelalearning.com\/osprecalc\/?post_type=chapter&#038;p=11202"},"modified":"2021-11-15T01:58:09","modified_gmt":"2021-11-15T01:58:09","slug":"introduction-to-logarithmic-properties","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/introduction-to-logarithmic-properties\/","title":{"raw":"Introduction to Logarithmic Properties","rendered":"Introduction to Logarithmic Properties"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>LEARNING OBJECTIVES<\/h3>\r\nBy the end of this lesson, you will be able to:\r\n<ul>\r\n \t<li>Use the product rule for logarithms.<\/li>\r\n \t<li>Use the quotient rule for logarithms.<\/li>\r\n \t<li>Use the power rule for logarithms.<\/li>\r\n \t<li>Expand logarithmic expressions.<\/li>\r\n \t<li>Condense logarithmic expressions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<figure id=\"CNX_Precalc_Figure_04_05_001\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" 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\" data-media-type=\"image\/jpg\" \/> <b>Figure 1.<\/b> The pH of hydrochloric acid is tested with litmus paper. (credit: David Berardan)[\/caption]<\/figure>\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;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}\\text{pH}=-\\mathrm{log}\\left(\\left[{H}^{+}\\right]\\right)\\hfill \\\\ \\text{ }=\\mathrm{log}\\left(\\frac{1}{\\left[{H}^{+}\\right]}\\right)\\hfill \\end{cases}[\/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\nThe following video gives a quick overview of the properties. Make sure to keep reading the text for more examples and detailed explanations.\r\n\r\n[embed]https:\/\/www.youtube.com\/watch?v=SxF44olWTyk[\/embed]\r\n\r\n<section id=\"fs-id1165137737533\" data-depth=\"1\"><\/section>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>LEARNING OBJECTIVES<\/h3>\n<p>By the end of this lesson, you will be able to:<\/p>\n<ul>\n<li>Use the product rule for logarithms.<\/li>\n<li>Use the quotient rule for logarithms.<\/li>\n<li>Use the power rule for logarithms.<\/li>\n<li>Expand logarithmic expressions.<\/li>\n<li>Condense logarithmic expressions.<\/li>\n<\/ul>\n<\/div>\n<figure id=\"CNX_Precalc_Figure_04_05_001\" class=\"small\">\n<div style=\"width: 254px\" class=\"wp-caption aligncenter\"><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\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1.<\/b> The pH of hydrochloric acid is tested with litmus paper. (credit: David Berardan)<\/p>\n<\/div>\n<\/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;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}\\text{pH}=-\\mathrm{log}\\left(\\left[{H}^{+}\\right]\\right)\\hfill \\\\ \\text{ }=\\mathrm{log}\\left(\\frac{1}{\\left[{H}^{+}\\right]}\\right)\\hfill \\end{cases}[\/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<p>The following video gives a quick overview of the properties. Make sure to keep reading the text for more examples and detailed explanations.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"The Properties of Logarithms\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/SxF44olWTyk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<section id=\"fs-id1165137737533\" data-depth=\"1\"><\/section>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-11202\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-11202","chapter","type-chapter","status-publish","hentry"],"part":11222,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/11202","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":9,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/11202\/revisions"}],"predecessor-version":[{"id":16400,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/11202\/revisions\/16400"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/parts\/11222"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/11202\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/media?parent=11202"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapter-type?post=11202"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/contributor?post=11202"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/license?post=11202"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}