{"id":2236,"date":"2024-07-15T21:48:11","date_gmt":"2024-07-15T21:48:11","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2236"},"modified":"2024-12-02T14:36:17","modified_gmt":"2024-12-02T14:36:17","slug":"logarithmic-properties-apply-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/logarithmic-properties-apply-it-1\/","title":{"raw":"Logarithmic Properties: Apply It 1","rendered":"Logarithmic Properties: Apply It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Use the basic properties of logarithms to simplify expressions and solve equations<\/li>\r\n \t<li>Combine or separate logarithms using the product and quotient rules<\/li>\r\n \t<li>Use the power rule to simplify logarithms with exponents<\/li>\r\n \t<li>Break down or combine complicated logarithm expressions into simpler forms<\/li>\r\n \t<li>Use the change-of-base formula to calculate and simplify logarithms with different bases<\/li>\r\n<\/ul>\r\n<\/section>In chemistry, pH is a measure of how acidic or basic a liquid is. It is essentially a measure of the concentration of hydrogen ions in a solution. The scale for measuring pH is standardized across the world, the scientific community having agreed upon its values and methods for acquiring them.\r\n\r\nMeasurements of pH can help scientists, farmers, doctors, and engineers solve problems and identify sources of problems.\r\n<p style=\"text-align: center;\">pH is defined as the decimal logarithm of the reciprocal of the hydrogen ion activity, [latex]a_{H}+[\/latex], in a solution.\r\n[latex]\\text{pH} =-\\log _{10}(a_{{\\text{H}}^{+}})=\\log _{10}\\left({\\frac {1}{a_{{\\text{H}}^{+}}}}\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center;\">For example, a solution with a hydrogen ion activity of [latex]2.5\u00d7{10}^{-6}[\/latex] (at that level essentially the number of moles of hydrogen ions per liter of solution) has a pH of [latex]\\log_{10}\\left(\\frac{1}{2.5\u00d7{10}^{-6}}\\right)=5.6[\/latex]<\/p>\r\n<p style=\"text-align: left;\">In the next examples, we will solve some problems involving pH.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Recall that, in chemistry, [latex]\\text{pH}=-\\mathrm{log}\\left[{H}^{+}\\right][\/latex]. If the concentration of hydrogen ions in a liquid is doubled, what is the effect on pH?[reveal-answer q=\"92345\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"92345\"]Suppose [latex]C[\/latex]\u00a0is the original concentration of hydrogen ions and [latex]P[\/latex]\u00a0is the original pH of the liquid. Then [latex]\\text{P}=-\\mathrm{log}\\left(C\\right)[\/latex]. If the concentration is doubled, the new concentration is [latex]2C[\/latex]. Then the pH of the new liquid is\r\n\r\n<center>[latex]\\text{pH}=-\\mathrm{log}\\left(2C\\right)[\/latex]<\/center>Using the product rule of logs\r\n\r\n<center>[latex]\\text{pH}=-\\mathrm{log}\\left(2C\\right)=-\\left(\\mathrm{log}\\left(2\\right)+\\mathrm{log}\\left(C\\right)\\right)=-\\mathrm{log}\\left(2\\right)-\\mathrm{log}\\left(C\\right)[\/latex]<\/center>Since [latex]P=-\\mathrm{log}\\left(C\\right)[\/latex], the new pH is\r\n\r\n<center>[latex]\\text{pH}=P-\\mathrm{log}\\left(2\\right)\\approx P - 0.301[\/latex]<\/center>When the concentration of hydrogen ions is doubled, the pH decreases by about [latex]0.301[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">How does the pH change when the concentration of positive hydrogen ions is decreased by half?[reveal-answer q=\"569571\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"569571\"]The pH increases by about [latex]0.301[\/latex].[\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question]293718[\/ohm_question]<\/section><section aria-label=\"Try It\"><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]293719[\/ohm_question]\r\n\r\n<\/section><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Use the basic properties of logarithms to simplify expressions and solve equations<\/li>\n<li>Combine or separate logarithms using the product and quotient rules<\/li>\n<li>Use the power rule to simplify logarithms with exponents<\/li>\n<li>Break down or combine complicated logarithm expressions into simpler forms<\/li>\n<li>Use the change-of-base formula to calculate and simplify logarithms with different bases<\/li>\n<\/ul>\n<\/section>\n<p>In chemistry, pH is a measure of how acidic or basic a liquid is. It is essentially a measure of the concentration of hydrogen ions in a solution. The scale for measuring pH is standardized across the world, the scientific community having agreed upon its values and methods for acquiring them.<\/p>\n<p>Measurements of pH can help scientists, farmers, doctors, and engineers solve problems and identify sources of problems.<\/p>\n<p style=\"text-align: center;\">pH is defined as the decimal logarithm of the reciprocal of the hydrogen ion activity, [latex]a_{H}+[\/latex], in a solution.<br \/>\n[latex]\\text{pH} =-\\log _{10}(a_{{\\text{H}}^{+}})=\\log _{10}\\left({\\frac {1}{a_{{\\text{H}}^{+}}}}\\right)[\/latex]<\/p>\n<p style=\"text-align: center;\">For example, a solution with a hydrogen ion activity of [latex]2.5\u00d7{10}^{-6}[\/latex] (at that level essentially the number of moles of hydrogen ions per liter of solution) has a pH of [latex]\\log_{10}\\left(\\frac{1}{2.5\u00d7{10}^{-6}}\\right)=5.6[\/latex]<\/p>\n<p style=\"text-align: left;\">In the next examples, we will solve some problems involving pH.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Recall that, in chemistry, [latex]\\text{pH}=-\\mathrm{log}\\left[{H}^{+}\\right][\/latex]. If the concentration of hydrogen ions in a liquid is doubled, what is the effect on pH?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q92345\">Show Solution<\/button><\/p>\n<div id=\"q92345\" class=\"hidden-answer\" style=\"display: none\">Suppose [latex]C[\/latex]\u00a0is the original concentration of hydrogen ions and [latex]P[\/latex]\u00a0is the original pH of the liquid. Then [latex]\\text{P}=-\\mathrm{log}\\left(C\\right)[\/latex]. If the concentration is doubled, the new concentration is [latex]2C[\/latex]. Then the pH of the new liquid is<\/p>\n<div style=\"text-align: center;\">[latex]\\text{pH}=-\\mathrm{log}\\left(2C\\right)[\/latex]<\/div>\n<p>Using the product rule of logs<\/p>\n<div style=\"text-align: center;\">[latex]\\text{pH}=-\\mathrm{log}\\left(2C\\right)=-\\left(\\mathrm{log}\\left(2\\right)+\\mathrm{log}\\left(C\\right)\\right)=-\\mathrm{log}\\left(2\\right)-\\mathrm{log}\\left(C\\right)[\/latex]<\/div>\n<p>Since [latex]P=-\\mathrm{log}\\left(C\\right)[\/latex], the new pH is<\/p>\n<div style=\"text-align: center;\">[latex]\\text{pH}=P-\\mathrm{log}\\left(2\\right)\\approx P - 0.301[\/latex]<\/div>\n<p>When the concentration of hydrogen ions is doubled, the pH decreases by about [latex]0.301[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">How does the pH change when the concentration of positive hydrogen ions is decreased by half?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q569571\">Show Solution<\/button><\/p>\n<div id=\"q569571\" class=\"hidden-answer\" style=\"display: none\">The pH increases by about [latex]0.301[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm293718\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=293718&theme=lumen&iframe_resize_id=ohm293718&source=tnh&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section aria-label=\"Try It\">\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm293719\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=293719&theme=lumen&iframe_resize_id=ohm293719&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<\/section>\n","protected":false},"author":12,"menu_order":9,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":280,"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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2236"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2236\/revisions"}],"predecessor-version":[{"id":6507,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2236\/revisions\/6507"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/280"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2236\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2236"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2236"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2236"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2236"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}