{"id":71935,"date":"2021-04-26T14:17:21","date_gmt":"2021-04-26T19:17:21","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=71935"},"modified":"2026-03-28T11:33:45","modified_gmt":"2026-03-28T16:33:45","slug":"logarithmic-functions","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/logarithmic-functions\/","title":{"rendered":"Logarithmic Function"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_vhALg-6wohY\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_vhALg-6wohY\" data-source-videoID=\"vhALg-6wohY\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Logarithmic Function Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Logarithmic Function\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_vhALg-6wohY:hover {cursor:pointer;} img#videoThumbnailImage_vhALg-6wohY {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/07\/updated-logarithmic-function-64c3dfefb0e82-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_vhALg-6wohY\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_vhALg-6wohY\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Logarithmic Function\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_vhALg-6wohY\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_vhALg-6wohY\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_vhALg-6wohY\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction LWK_Function() {\n  var x = document.getElementById(\"LWK\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"LWK_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"LWK\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Exponential_Equation_Basics\" class=\"smooth-scroll\">Exponential Equation Basics<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Logarithm_Notation\" class=\"smooth-scroll\">Logarithm Notation<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#The_Richter_Scale\" class=\"smooth-scroll\">The Richter Scale<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Graphing_Logarithmic_Functions\" class=\"smooth-scroll\">Graphing Logarithmic Functions<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Logarithmic_Function_Practice_Questions\" class=\"smooth-scroll\">Logarithmic Function Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hi, and welcome to this review of logarithmic functions!<\/p>\n<p>Like all mathematical functions, logarithms are used to measure and model real-life occurrences. From sound measured in decibels to the magnitude of earthquakes measured on the Richter Scale, logarithms are used to relate these natural occurrences to a baseline measurement.<\/p>\n<p>In addition, logarithmic functions have an inverse relationship with exponential functions, meaning that they \u201cundo\u201d each other. This allows us to use  logarithms as a tool to solve exponential equations.<\/p>\n<p>If you want to try some of the examples in this video on your own, grab a scientific calculator and be prepared to practice some skills and explore the power and properties of logarithmic functions.<\/p>\n<p>Let\u2019s get started!<\/p>\n<h2><span id=\"Exponential_Equation_Basics\" class=\"m-toc-anchor\"><\/span>Exponential Equation Basics<\/h2>\n<p>\nBefore we start our work with logarithms, let\u2019s take some time to review the basics of exponential equations.<\/p>\n<p>A base of 2 raised to the power of 3 is equal to 8. The answer, 8, is referred to as the argument. Rearranging these components allows us to write the inverse logarithm, as shown highlighted in green. This illustrates nicely that logs are the power of a named base.<\/p>\n<h3><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h3>\n<p>\nLet\u2019s practice identifying these components and re-writing a few exponential equations as logarithmic equations:<\/p>\n<div class=\"examplesentence\">\\(5^2=25\\)<\/div>\n<p>\n&nbsp;<br \/>\nNormally we would say, \u201cfive squared equals 25.\u201d If we were to use the terms we used in our first example, we could say, \u201ca base of 5 raised to the power of 2 is equal to 25.\u201d<\/p>\n<p>Now, let\u2019s rewrite this as a logarithm:<\/p>\n<div class=\"examplesentence\">\\(\\text{log}_525=2\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h3><span id=\"Example_2\" class=\"m-toc-anchor\"><\/span>Example #2<\/h3>\n<p>\nLet\u2019s try another one. If you\u2019d like to try this out on your own, pause the video and give it a shot!<\/p>\n<div class=\"examplesentence\">\\(4^3=64\\)<\/div>\n<p>\n&nbsp;<br \/>\nWe would say this is a base of 4 raised to the power of 3 is equal to 64.<\/p>\n<p>If we rewrite this as a logarithm, we get this:<\/p>\n<div class=\"examplesentence\">\\(\\text{log}_464=3\\)<\/div>\n<p>\n&nbsp;<br \/>\nWhen we evaluate a logarithmic expression we ask ourselves, \u201cWhat do I have to raise this base by to get the value of the argument?\u201d Logarithms are the power that is needed.<\/p>\n<h2><span id=\"Logarithm_Notation\" class=\"m-toc-anchor\"><\/span>Logarithm Notation<\/h2>\n<p>\nNow, let\u2019s focus on the notation of logarithms.<\/p>\n<p>In our first example, the base of the log was 5, and our second example had a base of 4. Note that the base is indicated as a subscript on the word, \u201clog.\u201d This will be true for logs with bases other than 10 and the irrational number, \\(e\\). <\/p>\n<p>Logarithms with a base of 10 do not indicate the base in the notation, and they are called \u201ccommon logs.\u201d There is a key on a scientific calculator for common logs. If you have a calculator on hand, try inputting the following: \\(\\text{log } 1000\\).<\/p>\n<p>The answer is 3, which means that 3 is the exponent needed to raise a base of 10 to get to 1,000. See if you can guess the answer to this one before you see the answer: \\(\\text{log } 10\\).<\/p>\n<p>Think you got it? The answer is 1. A base of 10 raised to the power of 1 will result in 10.<\/p>\n<h3><span id=\"Natural_Log\" class=\"m-toc-anchor\"><\/span>Natural Log<\/h3>\n<p>\nThere is also a key for the \u201cnatural\u201d log function, which reads, \u201cln.\u201d Whenever you see this notation, you know that you are dealing with a base of \\(e\\). Exponential functions that deal with continuously compounded interest or population growth models have a base of \\(e\\). The natural log is used to solve such applications, but that\u2019s a topic for another time.<\/p>\n<h2><span id=\"The_Richter_Scale\" class=\"m-toc-anchor\"><\/span>The Richter Scale<\/h2>\n<p>\nAs mentioned earlier, the Richter scale measures the amplitude of waves that result from seismic activity and uses a logarithmic formula \\(R=log(\\frac{A}{B})\\), where \\(A\\) is the observed amplitude and \\(B\\) is a baseline measure. <\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/02\/uhg-2.webp\" alt=\"\" width=\"\" height=\"\" class=\"aligncenter size-full wp-image-215971\"  role=\"img\" style=\"box-shadow: 1.5px 1.5px 3px gray;\"  \/><\/p>\n<p>If the amplitude of a wavelength is measured at 450 times the baseline, the formula would read as \\(R=\\text{log }(\\frac{450\\times B}{B})\\), which simplifies to \\(R=\\text{log }(450)\\). Entering this in your calculator results in a value of 2.7, rounded to the nearest tenth. This measurement reflects the exponent on a base 10, and would be considered a fairly minor earthquake on the Richter Scale.<\/p>\n<h2><span id=\"Graphing_Logarithmic_Functions\" class=\"m-toc-anchor\"><\/span>Graphing Logarithmic Functions<\/h2>\n<p>\nLike all inverse functions, the graphs of exponential and logarithmic functions are symmetric about the identity line \\(y = x\\).<\/p>\n<p>Graphing a logarithmic function is as simple as creating a table and plugging in values until the curve takes shape. <\/p>\n<p>Some students find it is easier to convert the logarithm to an exponential function and \u201cflip\u201d the domain and range values to get the mirror image graph.<\/p>\n<p>Let\u2019s give that method a try:<\/p>\n<p>Graph the exponential function \\(f(x) = 3^x\\) and the logarithmic function \\(g(x)=\\text{log}_3(x)\\) on the same graph.<\/p>\n<p>There are several properties of logarithms that allow you to rewrite expressions in order to simplify them. I will list the corresponding exponent rule that you are already familiar with that supports the logarithmic function. Given what we have already covered with respect to exponential expressions, the first three should seem rather intuitive:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/02\/uhg.webp\" alt=\"\" width=\"\" height=\"\" class=\"aligncenter size-full wp-image-215971\"  role=\"img\" style=\"box-shadow: 1.5px 1.5px 3px gray;\"  \/><\/p>\n<h3><span id=\"Property_1\" class=\"m-toc-anchor\"><\/span>Property #1<\/h3>\n<div class=\"examplesentence\">\\(\\text{log}_b1=0\\)<\/div>\n<p>\n&nbsp;<br \/>\nRemember to ask yourself, \u201cWhat base of \\(b^0\\) will give me an answer of 1?\u201d Well, we know that any base raised to a power of 0 equals 1.<\/p>\n<h3><span id=\"Property_2\" class=\"m-toc-anchor\"><\/span>Property #2<\/h3>\n<div class=\"examplesentence\">\\(\\text{log}_b(b)=1\\)<\/div>\n<p>\n&nbsp;<br \/>\nThis also makes sense, because raising any base to a power of 1 is equal to the base.<\/p>\n<h3><span id=\"Property_3\" class=\"m-toc-anchor\"><\/span>Property #3<\/h3>\n<div class=\"examplesentence\">\\(\\text{log}_b(b^n)=n\\)<\/div>\n<p>\n&nbsp;<br \/>\nThis is similar to Property #2, but the exponent, \\(n\\), can be any value. Thinking about the inverse relationship with an exponential also may be helpful here: taking the log of an exponential \u201cundoes\u201d the exponential. The result is the input, \\(n\\).<\/p>\n<p>The next three properties are more arithmetic and are used to simplify or expand logarithmic expressions. I will provide some examples for each of these properties.<\/p>\n<h3>Property #4 &#8211; The Product Property<\/h3>\n<div class=\"examplesentence\">\\(\\text{log}_b(m \\times n)=\\text{log}_b(m)+\\text{log}_b(n)\\)<\/div>\n<p>\n&nbsp;<br \/>\nThe key with this property is that you can \u201cexpand\u201d the log of a product to the sum of logs with the same base. This relates to the exponent rule.<\/p>\n<p>Here is an example:<\/p>\n<p>Write the logarithm as a sum and simplify, if possible.        <\/p>\n<div class=\"examplesentence\">\\(\\text{log}_4(24)\\rightarrow \\text{log}_4(4 \\times 6)=\\text{log}_4(4) + \\text{log}_4(6)\\)<\/div>\n<p>\n&nbsp;<br \/>\nFor this, we\u2019ll use Property #2 to simplify the first term of the expression on the right:<\/p>\n<div class=\"examplesentence\">\\(\\text{log}_4(24)=1+\\text{log}_4(6)\\)<\/div>\n<p>\n&nbsp;<br \/>\nHere\u2019s another example. Write the logarithm as a sum and simplify, if possible.<\/p>\n<div class=\"examplesentence\">\\(\\text{log}_3(9x)\\)<br \/>\n\\(\\text{log}_3(9) + \\text{log}_3(x)\\)<\/div>\n<p>\n&nbsp;<br \/>\nTo simplify it, let\u2019s rewrite \\(3^2\\).         <\/p>\n<div class=\"examplesentence\">\\(\\text{log}_3(3^2)+\\text{log}_3(x)\\)<\/div>\n<p>\n&nbsp;<br \/>\nThen, we\u2019ll use Property #3 to simplify further. This gives us:<\/p>\n<div class=\"examplesentence\">\\(2+\\text{log}_3(x)\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>Let\u2019s take a look at another property.<\/p>\n<h3>Property #5 &#8211; The Quotient Property<\/h3>\n<div class=\"examplesentence\">\\(\\text{log}_b(\\frac{m}{n})=\\text{log}_b(m)-\\text{log}_b(n)\\)<\/div>\n<p>\n&nbsp;<br \/>\nAs you can see, the log of a quotient can be expanded into the subtraction of two logs of the same base. The associated exponent rule is \\(\\frac{a^m}{a^n}=a^{m-n}\\).<\/p>\n<p>Let\u2019s practice expanding a logarithm using this property.<\/p>\n<div class=\"examplesentence\">\\(\\text{log}_7(\\frac{49}{3x})=\\text{log}_7(49)-\\text{log}_7(3x)\\)<\/div>\n<p>\n&nbsp;<br \/>\nWe need to be sure to keep the bases the same for the subtracted terms! The first term of the expression on the right can be simplified by writing 49 as \\(7^2\\).<\/p>\n<div class=\"examplesentence\">\\(\\text{log}_7(\\frac{49}{3x})=\\text{log}_7(7^2)-\\text{log}_7(3x)\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow, we can simplify with Property #3. This gives us:<\/p>\n<div class=\"examplesentence\">\\(\\text{log}_7(\\frac{49}{3x})=2-\\text{log}_7(3x)\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow let\u2019s work on the property backwards to condense the subtraction of logs with the same base to a quotient:<\/p>\n<div class=\"examplesentence\">\\(\\text{log}_5(9)-\\text{log}_5(3)\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow we\u2019re gonna create the quotient.<\/p>\n<div class=\"examplesentence\">\\(\\text{log}_5(\\frac{9}{3})\\)<\/div>\n<p>\n&nbsp;<br \/>\nSimplify the quotient.<\/p>\n<div class=\"examplesentence\">\\(\\text{log}_5(3)\\)<\/div>\n<p>\n&nbsp;<br \/>\nYou have condensed the expression to one log!<\/p>\n<p>The next property involves the exponent on the argument of a logarithm.<\/p>\n<h3>Property #6 &#8211; The Power Property<\/h3>\n<div class=\"examplesentence\">\\(\\text{log}_a(x)^p=p \\times \\text{log}_a(x)\\)<\/div>\n<p>\n&nbsp;<br \/>\nNote that when the argument is raised to a power, the expression is equal to the exponent being multiplied by the logarithm. The associated exponent rule is \\((a^m)^n=a^{m \\times n}\\).<\/p>\n<p>We will practice implementing this property with a more comprehensive example just to keep things interesting! <\/p>\n<div class=\"examplesentence\">\\(\\text{log}(2) + 3\\text{log}(3) &#8211; \\text{log}(9)\\)<\/div>\n<p>\n&nbsp;<br \/>\nNotice that there is no subscript indicated! What is the base? Remember, if there is no subscript indicated, the base is always 10. <\/p>\n<p>Now, we need to identify the Power Property in the second term. The multiplier of the 3 before the log translates to the argument of 3 being raised to the third power. So this is the same as saying, <\/p>\n<div class=\"examplesentence\">\\(\\text{log}(2) + \\text{log}(3^3) &#8211; \\text{log}(9)\\)<\/div>\n<p>\n&nbsp;<br \/>\nUse the Product Rule to combine the sum of \\(\\text{log}(2)\\) and \\(\\text{log}(3^3)\\).<\/p>\n<div class=\"examplesentence\">\\(\\text{log}(2 \\times 3^3)-\\text{log}(9)\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow, use the Quotient Rule to condense the subtraction of common logs to a single common log.<\/p>\n<div class=\"examplesentence\">\\(\\text{log}(\\frac{2\\times 3^3}{9})\\)<\/div>\n<p>\n&nbsp;<br \/>\nSimplify the quotient by canceling out the common factor of 9, for a final answer of <\/p>\n<div class=\"examplesentence\">\\(\\text{log}(2 \\times 3) = \\text{log}(6)\\)<\/div>\n<p>\n&nbsp;<br \/>\nFor our final practice problem, let\u2019s go the other way and expand a single log into an expression of addition and subtraction:<\/p>\n<div class=\"examplesentence\">\\(\\text{log}_3(\\frac{3\\times 2}{5})\\)<\/div>\n<p>\n&nbsp;<br \/>\nFirst, we\u2019ll expand the quotient into subtraction of two logs, base 3.    <\/p>\n<div class=\"examplesentence\">\\(\\text{log}_3(3x^2)-\\text{log}_3(5)\\)<\/div>\n<p>\n&nbsp;<br \/>\nThen, we\u2019ll expand the product in the first term to the sum of logs, base 3.    <\/p>\n<div class=\"examplesentence\">\\(\\text{log}_3(3) + \\text{log}_3(x^2)-\\text{log}_3(5)\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow, we need to identify the Power Property in the second term. Move the exponent of 2 to the front of the log as a multiplier.    <\/p>\n<div class=\"examplesentence\">\\(\\text{log}_3(3) + 2\\text{log}_3(x) &#8211; \\text{log}_3(5)\\)<\/div>\n<p>\n&nbsp;<br \/>\nSimplify the first term using Property #2.        <\/p>\n<div class=\"examplesentence\">\\(1 + 2\\text{log}_3(x) &#8211; \\text{log}_3(5)\\)<\/div>\n<p>\n&nbsp;<\/p>\n<hr>\n<p>\nOkay, that about wraps things up! I hope that the examples that we worked through have helped you better understand logarithms, their relationship with exponential functions, and the properties that allow us to simplify complex expressions.<\/p>\n<p>Thanks for watching, and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Logarithmic_Function_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Logarithmic Function Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nRewrite the exponential equation \\(7^3=343\\) as a logarithmic equation.<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">\\(\\text{log}_3(343)=7\\)<\/div><div class=\"PQ\"  id=\"PQ-1-2\">\\(\\text{log}_3(7)=343\\)<\/div><div class=\"PQ\"  id=\"PQ-1-3\">\\(\\text{log}_7(3)=343\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-4\">\\(\\text{log}_7(343)=3\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>In a logarithmic equation, the subscript next to \u201clog\u201d represents the base. The portion in parentheses is called the argument. The number after the equal sign is the power of the named base. In the given exponential equation, the base is 7, the argument is 343, and the power is 3. Answer D correctly rearranges these components to form the equivalent logarithmic equation. <\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nFill in the blank:<\/p>\n<div class=\"yellow-math-quote\">Graphs of logarithmic functions and exponential functions are __________ in relation to the identity line, \\(y=x\\).<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">asymmetrical<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-2\">symmetrical<\/div><div class=\"PQ\"  id=\"PQ-2-3\">parallel<\/div><div class=\"PQ\"  id=\"PQ-2-4\">perpendicular<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Since logarithmic functions and exponential functions are inverses of each other, they are symmetrical about the identity line \\(y=x\\). This concept is illustrated in the graph below.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Logarithmic-function-graph-example.svg\" alt=\"Graph showing the exponential function y = 2^x in blue and its inverse, the logarithmic function y = log\u2082x, in red on an x-y coordinate plane.\" width=\"411.4\" height=\"431.8\" class=\"aligncenter size-full wp-image-275482\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nMark created a table for the exponential function \\(f(x)=2x\\) and plugged in corresponding values. His table is shown below. Mark\u2019s partner created a table for the logarithmic function \\(g(x)=\\text{log}_2x\\). Based on Mark\u2019s work, which table should his partner have created?<\/p>\n<p style=\"text-align: center; margin-bottom: 0.5em\">\\(f(x)=2^x\\)<\/p>\n<table class=\"ATable\" style=\"width: 50%; margin: auto;\">\n<thead>\n<tr>\n<th>\\(x\\)<\/th>\n<th>\\(f(x)\\)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"background-color: white\">\n<td>0<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\"><p style=\"margin-bottom: 0.5em;\">\\(g(x)=\\log_2x\\)<\/p>\r\n<table class=\"ATable\">\r\n<tbody>\r\n<tr style=\"background-color: white\">\r\n<td style=\"background-color: #FFCC00\">\\(x\\)<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr style=\"background-color: white\">\r\n<td style=\"background-color: #FFCC00\">\\(g(x)\\)<\/td>\r\n<td>8<\/td>\r\n<td>4<\/td>\r\n<td>2<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table><\/div><div class=\"PQ\"  id=\"PQ-3-2\"><p style=\"margin-bottom: 0.5em\">\\(g(x)=\\log_2x\\)<\/p>\r\n<table class=\"ATable\">\r\n<tbody>\r\n<tr style=\"background-color: white\">\r\n<td style=\"background-color: #FFCC00\">\\(x\\)<\/td>\r\n<td>3<\/td>\r\n<td>2<\/td>\r\n<td>1<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<tr style=\"background-color: white\">\r\n<td style=\"background-color: #FFCC00\">\\(g(x)\\)<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table><\/div><div class=\"PQ\"  id=\"PQ-3-3\"><p style=\"margin-bottom: 0.5em\">\\(g(x)=\\log_2x\\)<\/p>\r\n<table class=\"ATable\">\r\n<tbody>\r\n<tr style=\"background-color: white\">\r\n<td style=\"background-color: #FFCC00\">\\(x\\)<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr style=\"background-color: white\">\r\n<td style=\"background-color: #FFCC00\">\\(g(x)\\)<\/td>\r\n<td>3<\/td>\r\n<td>2<\/td>\r\n<td>1<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table><\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-4\"><p style=\"margin-bottom: 0.5em\">\\(g(x)=\\log_2x\\)<\/p>\r\n<table class=\"ATable\">\r\n<tbody>\r\n<tr style=\"background-color: white\">\r\n<td style=\"background-color: #FFCC00\">\\(x\\)<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr style=\"background-color: white\">\r\n<td style=\"background-color: #FFCC00\">\\(g(x)\\)<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>To convert a table for an exponential function into a logarithmic function, flip the domain and range values to get the mirror image graph.<\/p>\n<p>Since the values of \\(x\\) in the exponential function are 0, 1, 2, and 3, the values of \\(g(x)\\) in the logarithmic function must also be 0, 1, 2, and 3.<\/p>\n<p>Likewise, since the values of \\(f(x)\\) in the exponential function are 1, 2, 4, and 8, the values of \\(x\\) in the logarithmic function must also be 1, 2, 4, and 8.<\/p>\n<p>Therefore, answer D is correct.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWrite the logarithm \\(\\text{log}_3(27)\\) as a sum and simplify if possible. <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-4-1\">3<\/div><div class=\"PQ\"  id=\"PQ-4-2\">2<\/div><div class=\"PQ\"  id=\"PQ-4-3\">\\(1+\\log_3(3^2)\\)<\/div><div class=\"PQ\"  id=\"PQ-4-4\">\\(1+\\log_3(9)\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>First, use the product property to rewrite the logarithm as the sum of \\(\\log_3(3)\\) and \\(\\log_3(9)\\) since \\(3\\times9=27\\).<\/p>\n<p style=\"text-align: center\">\\(\\log_3(27)=\\log_3(3)+\\log_3(9)\\)<\/p>\n<p>Next, use the second property, \\(\\log_b(b)=1\\) to rewrite \\(\\log_3(3)\\) as 1.<\/p>\n<p style=\"text-align: center\">\\(1+\\log_3(9)\\)<\/p>\n<p>Then, rewrite \\(\\log_3(9)\\) as \\(\\log_3(3^2)\\) since \\(3^2=9\\).<\/p>\n<p style=\"text-align: center\">\\(1+\\log_3(3^2)\\)<\/p>\n<p>From here, use the third property, \\(\\log_b(b^n)=n\\), to rewrite \\(\\log_3(3^2)\\) as 2.<\/p>\n<p style=\"text-align: center\">\\(1+2=3\\)<\/p>\n<p>Since \\(1+2=3\\), the logarithm can be simplified to 3. Therefore, answer A is correct.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich property is illustrated below?<\/p>\n<div class=\"yellow-math-quote\" style=\"line-height: 45px\">\\(\\log_8(12)-\\log_8(4)\\)<br \/>\n\\(\\log_8(\\large{\\frac{12}{4}}\\normalsize{)}\\)<br \/>\n\\(\\log_8(3)\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">Identity property <\/div><div class=\"PQ\"  id=\"PQ-5-2\">Power property <\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-3\">Quotient property <\/div><div class=\"PQ\"  id=\"PQ-5-4\">Product property <\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>In the example given, the logarithmic expression is simplified using the quotient property. The quotient property is as follows:<\/p>\n<p style=\"text-align: center\">\\(\\log_b(\\large{\\frac{m}{n}}\\normalsize{)=\\log_b(m)-\\log_b(n)}\\)<\/p>\n<p>The example gave \\(\\log_b(m)-\\log_b(n)\\), which needed to be simplified to \\(\\log_b(\\frac{m}{n})\\).<\/p>\n<p>Therefore, Choice C is correct.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/algebra-ii\/\">Return to Algebra II Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Algebra II Videos<\/p>\n","protected":false},"author":1,"featured_media":187658,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-71935","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-miscellaneous-videos","7":"page_category-video-pages-for-study-course-sidebar-ad","8":"page_type-video","9":"content_type-practice-questions","10":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/71935","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=71935"}],"version-history":[{"count":7,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/71935\/revisions"}],"predecessor-version":[{"id":280973,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/71935\/revisions\/280973"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/187658"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=71935"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}