{"id":2533,"date":"2026-09-16T08:08:07","date_gmt":"2026-09-16T08:08:07","guid":{"rendered":"https:\/\/www.think10x.ai\/blog\/?p=2533"},"modified":"2026-09-16T08:08:09","modified_gmt":"2026-09-16T08:08:09","slug":"natural-logarithms-explained","status":"publish","type":"post","link":"https:\/\/www.think10x.ai\/blog\/natural-logarithms-explained\/","title":{"rendered":"Natural Logarithms Explained With Examples"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"2533\" class=\"elementor elementor-2533\" data-elementor-post-type=\"post\">\n\t\t\t\t<div class=\"elementor-element elementor-element-390dc4e e-flex e-con-boxed e-con e-parent\" data-id=\"390dc4e\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-d688a3f elementor-widget elementor-widget-text-editor\" data-id=\"d688a3f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">A natural logarithm is a logarithm with base e, the mathematical constant approximately equal to 2.71828. Written as ln(x), it answers the question &#8211; to what power must e be raised to get x<\/span><\/p><p><span style=\"font-weight: 400;\">Natural logarithms show up throughout calculus, finance, and science because e is the base that describes continuous growth, whether that&#8217;s compound interest, population growth, or radioactive decay. Once you understand that ln(x) asks &#8220;how long does it take to grow to x, at a continuous 100% growth rate,&#8221; most of the rules around it start to make sense rather than needing to be memorized.<\/span><\/p><h2>Key Takeaways<\/h2><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">A natural log, written ln(x), is the logarithm of x to base e, where e \u2248 2.71828.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">ln(x) and e\u02e3 are inverse functions, so ln(e\u02e3) = x and e^(ln x) = x.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The three main natural logarithm properties are the product rule, quotient rule, and power rule, and they work the same way as any other logarithm base.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The natural log is only defined for positive numbers. ln(0) is undefined, and the natural log of a negative number has no real solution.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">ln(1) = 0 and ln(e) = 1 are two fixed reference points worth memorizing, since they anchor every other calculation.<\/span><\/li><\/ul><h2>Natural Logarithm Explained Simply<\/h2><p><span style=\"font-weight: 400;\">Natural logarithm is the logarithm of a number to the base e, where e is an irrational constant approximately equal to 2.71828. It is written as ln(x) instead of log(x), and the &#8220;e&#8221; base is implied rather than written out. So ln(x) and log\u2091(x) mean exactly the same thing.<\/span><\/p><p><span style=\"font-weight: 400;\">The reason e shows up so often is that it describes continuous growth or decay, the kind that happens smoothly rather than in discrete jumps. Compound interest calculated continuously, radioactive decay, and population models under ideal conditions are all naturally described using e, which is why the logarithm built around that same base earns the name &#8220;natural.&#8221;\u00a0<\/span><\/p><p><a href=\"https:\/\/www.think10x.ai\/blog\/ai-explainer-video-tool\/\"><i><span style=\"font-weight: 400;\">See how a narrated video breaks down math concepts step by step<\/span><\/i><\/a><i><span style=\"font-weight: 400;\">.<\/span><\/i><\/p><h2>Natural Logarithm Formula<\/h2><p><span style=\"font-weight: 400;\">The natural logarithm formula defines ln(x) as the inverse of the exponential function e\u02e3 &#8211;<\/span><\/p><p><b>If e\u02b8 = x, then ln(x) = y<\/b><\/p><p><span style=\"font-weight: 400;\">This means ln(x) answers the question: what power do I need to raise e to in order to get x? For example, since e\u00b2 \u2248 7.389, we know ln(7.389) \u2248 2.<\/span><\/p><p><span style=\"font-weight: 400;\">According to the <\/span><a href=\"https:\/\/dlmf.nist.gov\/4.8\" rel=\"noopener\"><span style=\"font-weight: 400;\">NIST Digital Library of Mathematical Functions<\/span><\/a><span style=\"font-weight: 400;\">, the natural logarithm is defined for all positive real numbers, and its properties as an elementary function connect directly to the identities that govern logarithms of any base, since ln is simply the specific case where the base equals e.<\/span><\/p><h2>Natural Logarithm Properties<\/h2><p><span style=\"font-weight: 400;\">The core natural logarithm properties follow the same pattern as logarithms of any base, applied specifically with base e.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-97f9ed1 e-flex e-con-boxed e-con e-parent\" data-id=\"97f9ed1\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-4044145 eael-table-align-center eael-dt-th-align-left elementor-widget elementor-widget-eael-data-table\" data-id=\"4044145\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"eael-data-table.default\">\n\t\t\t\t\t\t\t<div class=\"eael-data-table-wrap\" data-table_id=\"4044145\" id=\"eael-data-table-wrapper-4044145\" data-custom_responsive=\"false\">\n\t\t\t<table class=\"tablesorter eael-data-table center\" id=\"eael-data-table-4044145\">\n\t\t\t    <thead>\n\t\t\t        <tr class=\"table-header\">\n\t\t\t\t\t\t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">Property<\/span><\/th>\n\t\t\t        \t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">Rule<\/span><\/th>\n\t\t\t        \t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">Example<\/span><\/th>\n\t\t\t        \t\t\t\t        <\/tr>\n\t\t\t    <\/thead>\n\t\t\t  \t<tbody>\n\t\t\t\t\t\t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tProduct Rule\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tln(ab) = ln(a) + ln(b)\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tln(6) = ln(2) + ln(3)\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tQuotient Rule\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tln(a\/b) = ln(a) \u2212 ln(b)\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tln(5) = ln(10) \u2212 ln(2)\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tPower Rule\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tln(a\u207f) = n \u00b7 ln(a)\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tln(8) = ln(2\u00b3) = 3ln(2)\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tIdentity of 1\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tIn(1) = 0\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tAlways true\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tIdentity of e\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tln(e) = 1\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tAlways true\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tInverse Property\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tln(e\u02e3) = x and e^(ln x) = x\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tln(e\u2075) = 5\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t    <\/tbody>\n\t\t\t<\/table>\n\t\t<\/div>\n\t  \t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-12d9e88 e-flex e-con-boxed e-con e-parent\" data-id=\"12d9e88\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-2d1aa37 elementor-widget elementor-widget-text-editor\" data-id=\"2d1aa37\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<h2>How to Calculate Natural Logarithm<\/h2><p><span style=\"font-weight: 400;\">Below are ways to calculate the natural logarithm<\/span><\/p><ol><li style=\"font-weight: 400;\" aria-level=\"1\"><b>With a calculator &#8211;<\/b><span style=\"font-weight: 400;\">\u00a0Most scientific and graphing calculators have a dedicated &#8220;ln&#8221; button. Enter the number and press ln to get the result directly.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Using known values &#8211;<\/b><span style=\"font-weight: 400;\">\u00a0Memorize a few reference points, ln(1) = 0, ln(e) = 1, ln(e\u00b2) \u2248 2, and build outward from there using the product, quotient, and power rules.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Breaking down composite numbers &#8211;<\/b><span style=\"font-weight: 400;\">\u00a0For a number like ln(12), rewrite it as ln(4 \u00d7 3) = ln(4) + ln(3), then look up or calculate each smaller piece.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Using the change of base formula, if needed &#8211;<\/b><span style=\"font-weight: 400;\">\u00a0ln(x) = log(x) \/ log(e), useful if you only have a base-10 log calculator available.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Upload a natural log problem to Think10x.ai<\/b><span style=\"font-weight: 400;\"> and <\/span><a href=\"https:\/\/www.think10x.ai\/studio\"><span style=\"font-weight: 400;\">get a narrated video<\/span><\/a><span style=\"font-weight: 400;\"> walking through each calculation step.<\/span><\/li><\/ol><h2>How to Solve Natural Logarithms in Equations<\/h2><h3>Step-by-step walkthrough -&gt; Solve ln(x) = 3<\/h3><ol><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The equation already has ln(x) isolated on one side.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Apply e to both sides to undo the natural log &#8211; e^(ln x) = e\u00b3.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Since e^(ln x) = x, this simplifies to x = e\u00b3.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">x \u2248 20.09.<\/span><\/li><\/ol><h3>Step-by-step walkthrough -&gt; Solve ln(2x) \u2212 ln(4) = 0<\/h3><ol><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Combine the logs using the quotient rule. ln(2x\/4) = 0.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Simplify inside the log &#8211; ln(x\/2) = 0.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Since ln(1) = 0, this means x\/2 = 1.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">x = 2.<\/span><\/li><\/ol><h2>Natural Logarithm Examples<\/h2><p><span style=\"font-weight: 400;\">Here are natural logarithm examples covering the calculation types that show up most often in coursework.<\/span><\/p><h3>Example 1. Simplifying using properties<\/h3><p><span style=\"font-weight: 400;\">Simplify ln(20) \u2212 ln(4).<\/span><\/p><p><span style=\"font-weight: 400;\">Using the quotient rule &#8211; ln(20) \u2212 ln(4) = ln(20\/4) = ln(5).<\/span><\/p><h3>Example 2. Expanding a natural log expression<\/h3><p><span style=\"font-weight: 400;\">Expand ln(3x\u00b2).<\/span><\/p><p><span style=\"font-weight: 400;\">Using the product and power rules &#8211; ln(3x\u00b2) = ln(3) + ln(x\u00b2) = ln(3) + 2ln(x).<\/span><\/p><h3><b>Example 3. Solving a natural log equation<\/b><\/h3><p><span style=\"font-weight: 400;\">Solve ln(x + 1) = 2.<\/span><\/p><p><span style=\"font-weight: 400;\">Apply e to both sides = x + 1 = e\u00b2. Since e\u00b2 \u2248 7.389, x \u2248 6.389.<\/span><\/p><h3>Example 4. A continuous growth application<\/h3><p><span style=\"font-weight: 400;\">A bacteria culture grows according to the model N(t) = N\u2080e^(0.3t), where t is time in hours. How long until the culture doubles?<\/span><\/p><ul><li><span style=\"font-weight: 400;\">Set N(t) = 2N\u2080: 2N\u2080 = N\u2080e^(0.3t). <\/span><\/li><li><span style=\"font-weight: 400;\">Divide both sides by N\u2080: 2 = e^(0.3t). <\/span><\/li><li><span style=\"font-weight: 400;\">Take the natural log of both sides &#8211; ln(2) = 0.3t. <\/span><\/li><li><span style=\"font-weight: 400;\">Solve t = ln(2) \/ 0.3 \u2248 2.31 hours.<\/span><\/li><\/ul><p><a href=\"https:\/\/www.think10x.ai\/blog\/math-solver-online-chatgpt-vs-think10x-ai\/\"><i><span style=\"font-weight: 400;\">Get a full walkthrough for a calculus or algebra numerical<\/span><\/i><\/a><i><span style=\"font-weight: 400;\"> using Think10x.ai&#8217;s step-by-step solver.<\/span><\/i><\/p><h2>Natural Logarithm Function: Domain, Range, and Graph<\/h2><p><span style=\"font-weight: 400;\">The<\/span> <span style=\"font-weight: 400;\">natural logarithm function f(x) = ln(x) has a domain of all positive real numbers, (0, \u221e), and a range of all real numbers, (\u2212\u221e, \u221e). The graph passes through (1, 0) since ln(1) = 0, and it rises slowly as x increases, with a vertical asymptote at x = 0 because ln(x) approaches negative infinity as x approaches zero from the right.<\/span><\/p><h2>Test Your Understanding<\/h2><p><span style=\"font-weight: 400;\">Try answering these before checking the answers below, or<\/span><a href=\"https:\/\/www.think10x.ai\/video-library\"> <span style=\"font-weight: 400;\">browse more math explanations<\/span><\/a><span style=\"font-weight: 400;\"> in the Think10x.ai video library.<\/span><\/p><ol><li style=\"font-weight: 400;\" aria-level=\"1\"><b>What is ln(1)?<\/b><span style=\"font-weight: 400;\"> Answer is 0. Any base raised to the power of 0 equals 1, so ln(1) = 0 for the same reason log(1) = 0 in any base.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Simplify ln(x\u2075) using the power rule.<\/b><span style=\"font-weight: 400;\"> Answer is 5ln(x). The exponent moves out front as a multiplier.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Is ln(\u22125) defined?<\/b><span style=\"font-weight: 400;\"> Answer is No. The natural logarithm is only defined for positive real numbers, so the natural log of a negative number has no real solution.<\/span><\/li><\/ol><h2>How Think10x.ai Helps With Natural Logarithms<\/h2><p><span style=\"font-weight: 400;\">Natural logarithm questions get confusing fast once properties start stacking, expanding a log with three factors, solving an equation with logs on both sides, or applying the natural log to a real-world growth model. Think10x.ai turns a natural log question, typed, spoken, or photographed from a worksheet, into a narrated video that walks through which property applies at each step, so the reasoning stays visible instead of disappearing into a memorized shortcut.<\/span><\/p><p><a href=\"https:\/\/www.think10x.ai\/\"><span style=\"font-weight: 400;\">Try it with your next math problem<\/span><\/a><span style=\"font-weight: 400;\"> and see the full walkthrough in under two minutes.<\/span><\/p><h2>Frequently Asked Questions<\/h2>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-969bca7 e-flex e-con-boxed e-con e-parent\" data-id=\"969bca7\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-4e5e193 elementor-widget elementor-widget-eael-adv-accordion\" data-id=\"4e5e193\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"eael-adv-accordion.default\">\n\t\t\t\t\t            <div class=\"eael-adv-accordion\" id=\"eael-adv-accordion-4e5e193\" data-scroll-on-click=\"no\" data-scroll-speed=\"300\" data-accordion-id=\"4e5e193\" data-accordion-type=\"accordion\" data-toogle-speed=\"300\">\n            <div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"what-is-the-difference-between-ln-and-log\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"1\" aria-controls=\"elementor-tab-content-8211\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">What is the difference between ln and log?<\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-8211\" class=\"eael-accordion-content clearfix\" data-tab=\"1\" aria-labelledby=\"what-is-the-difference-between-ln-and-log\"><p><span style=\"font-weight: 400\">&#8220;log&#8221; by itself usually means base-10 logarithm, while &#8220;ln&#8221; always means base-e logarithm, the natural logarithm. Both work the same way mathematically, just with a different base.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"why-is-e-used-as-the-base-for-natural-logarithms-instead-of-10-\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"2\" aria-controls=\"elementor-tab-content-8212\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">Why is e used as the base for natural logarithms instead of 10? <\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-8212\" class=\"eael-accordion-content clearfix\" data-tab=\"2\" aria-labelledby=\"why-is-e-used-as-the-base-for-natural-logarithms-instead-of-10-\"><p><span style=\"font-weight: 400\">Because e is the unique number where the function e\u02e3 is its own derivative, making calculus involving growth, decay, and rates of change dramatically simpler when e is the base rather than 10 or any other number.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"can-natural-logarithms-be-negative-\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"3\" aria-controls=\"elementor-tab-content-8213\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">Can natural logarithms be negative? <\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-8213\" class=\"eael-accordion-content clearfix\" data-tab=\"3\" aria-labelledby=\"can-natural-logarithms-be-negative-\"><p><span style=\"font-weight: 400\">Yes, when the input is between 0 and 1. For example, ln(0.5) \u2248 \u22120.693, since e raised to a negative power gives a fraction less than 1.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"how-do-you-convert-a-natural-log-to-a-common-log-or-vice-versa\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"4\" aria-controls=\"elementor-tab-content-8214\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">How do you convert a natural log to a common log, or vice versa?<\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-8214\" class=\"eael-accordion-content clearfix\" data-tab=\"4\" aria-labelledby=\"how-do-you-convert-a-natural-log-to-a-common-log-or-vice-versa\"><p><span style=\"font-weight: 400\">Use the change of base formula: ln(x) = log(x) \/ log(e), or equivalently log(x) = ln(x) \/ ln(10). A scientific calculator can compute either directly if you only have one of the two buttons available.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"is-eln-x-always-equal-to-x\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"5\" aria-controls=\"elementor-tab-content-8215\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">Is e^(ln x) always equal to x?<\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-8215\" class=\"eael-accordion-content clearfix\" data-tab=\"5\" aria-labelledby=\"is-eln-x-always-equal-to-x\"><p><span style=\"font-weight: 400\">Yes, for any positive x. This is the defining inverse relationship between the exponential function and the natural logarithm, and it&#8217;s the key step for solving most natural log equations.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>A natural logarithm is a logarithm with base e, the mathematical constant approximately equal to 2.71828. Written as ln(x), it answers the question &#8211; to what power must e be raised to get x Natural logarithms show up throughout calculus, finance, and science because e is the base that describes continuous growth, whether that&#8217;s compound [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":2538,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"fifu_image_url":"https:\/\/media.mentomind.ai\/img\/t10x\/bp\/natural_logarithms_explained_with_examples.webp","fifu_image_alt":"","footnotes":""},"categories":[8,1],"tags":[],"class_list":["post-2533","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-blogs","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/posts\/2533","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/comments?post=2533"}],"version-history":[{"count":4,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/posts\/2533\/revisions"}],"predecessor-version":[{"id":2537,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/posts\/2533\/revisions\/2537"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/media\/2538"}],"wp:attachment":[{"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/media?parent=2533"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/categories?post=2533"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/tags?post=2533"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}