{"id":4005,"date":"2019-06-29T09:55:11","date_gmt":"2019-06-29T00:55:11","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=4005"},"modified":"2019-11-24T00:03:00","modified_gmt":"2019-11-23T15:03:00","slug":"calculus-analytic-definition-of-exponential-function","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-analytic-definition-of-exponential-function\/","title":{"rendered":"\uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \uc774\uc6a9\ud55c \uc9c0\uc218\ud568\uc218\uc758 \uc815\uc758"},"content":{"rendered":"<p>\uc911\ud559\uad50\uc640 \uace0\ub4f1\ud559\uad50 \uacfc\uc815\uc5d0\uc11c\ub294 \uac19\uc740 \uc218\ub97c \uc5ec\ub7ec \ubc88 \uacf1\ud55c \uac83\uc73c\ub85c \uac70\ub4ed\uc81c\uacf1\uc744 \uc815\uc758\ud55c \ub4a4 \uac70\ub4ed\uc81c\uacf1\uc758 \uc9c0\uc218\ub97c \uc815\uc218, \uc720\ub9ac\uc218 \ubc94\uc704\ub85c \ud655\uc7a5\ud558\uba70, \uadf9\ud55c\uc744 \uc774\uc6a9\ud558\uc5ec \uc2e4\uc218 \uc9c0\uc218\ub97c \uc815\uc758\ud55c\ub2e4. \uadf8 \ub4a4\uc5d0 \uc9c0\uc218\ub97c \ubcc0\uc218\ub85c \uac16\ub294 \ud568\uc218\ub97c \uc9c0\uc218\ud568\uc218\ub85c \uc815\uc758\ud558\uba70, \uc9c0\uc218\ud568\uc218\uc758 \uc5ed\ud568\uc218\ub97c \ub85c\uadf8\ud568\uc218\ub85c \uc815\uc758\ud55c\ub2e4. \uc774\uc640 \uac19\uc774 \uc815\uc758\ub41c \uc9c0\uc218\ud568\uc218\ub294 \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uac00 \ub418\uba70, \ud2b9\ud788 \ubc11\uc774 \uc790\uc5f0\uc9c0\uc218\uc778 \ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<br \/>\n\\[e^x = 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\frac{x^4}{4!} + \\cdots\\]<br \/>\n\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uc21c\uc11c\ub97c \ubc14\uafb8\uc5b4 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \uc9c0\uc218\ud568\uc218\ub97c \uc815\uc758\ud558\uace0, \uadf8\ub807\uac8c \uc815\uc758\ub41c \uc9c0\uc218\ud568\uc218\uac00 \uc6b0\ub9ac\uac00 \uc6d0\ud558\ub294 \uc131\uc9c8\uc744 \ubaa8\ub450 \uac00\uc9c0\uace0 \uc788\uc74c\uc744 \ubcf4\uc774\uaca0\ub2e4.<\/p>\n<div class=\"box contentindex\">\n<h3 class=\"indextitle\">\ub0b4\uc6a9 \uc21c\uc11c<\/h3>\n<ul>\n<li><a href=\"#analyticdefinition\">\uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \ud574\uc11d\uc801 \uc815\uc758<\/a><\/li>\n<li><a href=\"#property\">\uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \uc131\uc9c8<\/a><\/li>\n<li><a href=\"#naturallogarithm\">\uc790\uc5f0\ub85c\uadf8\ud568\uc218<\/a><\/li>\n<li><a href=\"#exponentialrule\">\uc9c0\uc218 \ubc95\uce59<\/a><\/li>\n<li><a href=\"#uniqueness\">\uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \uc720\uc77c\uc131<\/a><\/li>\n<li><a href=\"#generallogarithm\">\uc77c\ubc18\uc801\uc778 \ub85c\uadf8\ud568\uc218<\/a><\/li>\n<\/ul>\n<h3 class=\"pretitle\">\ubbf8\ub9ac \uc54c\uc544\uc57c \ud560 \ub0b4\uc6a9<\/h3>\n<ul>\n<li>\uac70\ub4ed\uc81c\uacf1\uae09\uc218 (<a href=\"https:\/\/en.wikipedia.org\/wiki\/Power_series\">\uad00\ub828 \uae00<\/a>)<\/li>\n<li>\ud14c\uc77c\ub7ec \uae09\uc218 (<a href=\"\/blog\/articles\/calculus-taylor-series-and-maclaurin-series\/\">\uad00\ub828 \uae00<\/a>)<\/li>\n<li>\uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218 (<a href=\"\/blog\/articles\/calculus-derivatives-of-exponential-and-logarithm-functions\/\">\uad00\ub828 \uae00<\/a>)<\/li>\n<\/ul>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"analyticdefinition\"><\/a><\/p>\n<h3>\uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \ud574\uc11d\uc801 \uc815\uc758<\/h3>\n<p>\uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \ud568\uc218\ub97c \uc815\uc758\ud558\ub294 \uac83\uc744 <span class=\"defined\">\ud574\uc11d\uc801\uc73c\ub85c \uc815\uc758\ud55c\ub2e4<\/span>\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc790\uc5f0\uc9c0\uc218\ud568\uc218\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \ud574\uc11d\uc801\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 1. (\uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \ud574\uc11d\uc801 \uc815\uc758)<\/span><\/p>\n<p>\uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\exp (x) = \\sum_{n=0}^{\\infty} \\frac{x^n}{n!} \\tag{1}\\]<br \/>\n\uc774 \ud568\uc218\ub97c <span class=\"defined\">\uc790\uc5f0\uc9c0\uc218\ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.\n<\/p>\n<\/div>\n<p>\ube44 \ud310\uc815\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec (1)\uc758 \uc218\ub834\ubc18\uc9c0\ub984\uc744 \uad6c\ud574 \ubcf4\uc790.<br \/>\n\\[\\lim_{n\\to\\infty} \\frac{n!}{(n+1)!} = 0\\]<br \/>\n\uc774\ubbc0\ub85c (1)\uc758 \uc218\ub834\ubc18\uc9c0\ub984\uc740 \ubb34\ud55c\ub300\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec (1)\uc740 \uc798 \uc815\uc758\ub41c \ud568\uc218\uc774\ub2e4.<\/p>\n<p><!-- ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"property\"><\/a><\/p>\n<h3>\uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \uc131\uc9c8<\/h3>\n<p>\uc774\uc81c (1)\uacfc \uac19\uc774 \uc815\uc758\ub41c \uc790\uc5f0\uc9c0\uc218\ud568\uc218\uac00 \uace0\ub4f1\ud559\uad50 \uacfc\uc815\uc5d0\uc11c \uc815\uc758\ud55c \uc9c0\uc218\ud568\uc218\uc758 \uc131\uc9c8\uc744 \ubaa8\ub450 \uac00\uc9c0\uace0 \uc788\uc74c\uc744 \ubcf4\uc774\uc790. \uc720\ub9ac\uc218 \uc9c0\uc218\uc5d0 \ub300\ud55c \uc9c0\uc218\ubc95\uce59\uc740 \uc774\ubbf8 \uc99d\uba85\ub418\uc5c8\ub2e4\uace0 \uac00\uc815\ud55c\ub2e4.<\/p>\n<p><!-- p>\uc73c\uba70, \ub610\ud55c \uadf8\ub7ec\ud55c \uc131\uc9c8\uc744 \uac16\uace0 \uc788\ub294 \ud568\uc218\uac00 \uc720\uc77c\ud568\uc73c\ub85c \ubcf4\uc784\uc73c\ub85c\uc368 \uc774 \uc815\uc758\ub97c \uc815\ub2f9\ud654\ud558\uc790.<\/p -->\n<p>\\(x\\)\uc640 \\(y\\)\uac00 \uc2e4\uc218\uc77c \ub54c, Mertens\uc758 \uc815\ub9ac(<a href=\"\/blog\/articles\/calculus-infinite-series\/\">\ucc38\uace0: \uc815\ub9ac 3<\/a>)\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\begin{align}<br \/>\n\\exp (x) \\exp (y)<br \/>\n&#038;= \\left( \\sum_{n=0}^{\\infty} \\frac{x^n}{n!} \\right) \\left( \\sum_{n=0}^{\\infty} \\frac{y^n}{n!} \\right) \\\\[3pt]<br \/>\n&#038;= \\sum_{n=0}^{\\infty} \\sum_{k=0}^{n} \\frac{x^k y^{n-k}}{k! (n-k)!} \\\\[3pt]<br \/>\n&#038;= \\sum_{n=0}^{\\infty} \\frac{1}{n!} \\sum_{k=0}^{n} \\frac{n! x^k y^{n-k}}{k! (n-k)!} \\\\[3pt]<br \/>\n&#038;= \\sum_{n=0}^{\\infty} \\frac{(x+y)^n}{n!} = \\exp(x+y)<br \/>\n\\end{align}\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\exp(x) \\exp (y) = \\exp (x+y)\\tag{2}\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4. (1)\uc5d0 \\(x=0\\)\uc744 \ub300\uc785\ud558\uba74 \\(\\exp(0) = 1\\)\uc774\ubbc0\ub85c<br \/>\n\\[\\exp (x) \\exp(-x) = \\exp(x-x) = \\exp(0) = 1\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(\\exp\\)\ub294 \uc5f0\uc18d\ud568\uc218\uc774\ubbc0\ub85c \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(\\exp (x) \\ne 0\\)\uc774\uace0<br \/>\n\\[\\exp(-x) = \\frac{1}{\\exp(x)}\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub610\ud55c \\(\\exp\\)\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uba74<br \/>\n\\[\\frac{d}{dx} \\exp (x) = \\sum_{n=1}^{\\infty} \\frac{nx^{n-1}}{n!} = \\sum_{n=0}^{\\infty} \\frac{x^n}{n!} = \\exp(x)\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(\\exp (0) = 1\\)\uc774\uace0 \\(\\exp\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc774\uba70 \\(\\exp(x) \\ne 0\\)\uc774\ubbc0\ub85c \uc0ac\uc787\uac12 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(\\exp (x) > 0\\)\uc774\ub2e4. \uc9c0\uae08\uae4c\uc9c0 \uc0b4\ud3b4\ubcf8 \ub0b4\uc6a9\uc744 \uc815\ub9ac\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 1. (\uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \uc131\uc9c8)<\/span><\/p>\n<p>\uc790\uc5f0\uc9c0\uc218\ud568\uc218 \\(\\exp\\)\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<ol class=\"bracket\">\n<li>\\(\\exp\\)\ub294 \\(\\mathbb{R}\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70 \uc99d\uac00\ud558\ub294 \ud568\uc218\uc774\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \uc2e4\uc218 \\(x,\\) \\(y\\)\uc5d0 \ub300\ud558\uc5ec \\(\\exp(x) \\exp(y) = \\exp(x+y)\\)\uc774\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(\\exp(-x) = [\\exp (x)]^{-1}\\)\uc774\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(\\frac{d}{dx} \\exp (x) = \\exp (x)\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\uc815\uc758 (1)\uacfc \\(e\\)\uc758 \uc131\uc9c8(<a href=\"\/blog\/articles\/calculus-e-and-pi\/\">\ucc38\uace0: \ubcf4\uc870\uc815\ub9ac 2<\/a>)\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\exp (1) = \\sum_{n=0}^{\\infty} \\frac{1}{n!}= e.\\tag{3}\\]<br \/>\n\ub530\ub77c\uc11c \\(n\\)\uc774 \uc790\uc5f0\uc218\uc77c \ub54c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\exp(n) = \\exp(1+1+ \\cdots + 1) = \\exp(1) \\exp(1) \\cdots \\exp(1) = ee \\cdots e = e^n .\\]<br \/>\n\ub610\ud55c \uc74c\uc758 \uc815\uc218 \\(-n\\)\uc5d0 \ub300\ud574\uc11c\ub294<br \/>\n\\[\\exp(-n) = \\frac{1}{\\exp(n)}= \\frac{1}{e^n} = e^{-n}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub9ac\uace0 \uc815\uc218 \\(m\\)\uacfc \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(r= \\frac{m}{n}\\)\uc774\ub77c\uace0 \ud588\uc744 \ub54c<br \/>\n\\[[\\exp(r)]^n = \\exp(r) \\exp(r) \\cdots \\exp(r) = \\exp(r+r+ \\cdots + r) = \\exp(nr) = e^{nr}\\]<br \/>\n\uc774\ubbc0\ub85c \\(\\exp(r) = \\left( e^{nr} \\right)^{1\/n} = e^{r}\\)\uc774\ub2e4. \ub530\ub77c\uc11c \uc784\uc758\uc758 \uc720\ub9ac\uc218 \\(r\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\exp(r) = e^r\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc774\uc81c \ubb34\ub9ac\uc218 \\(x\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(x\\)\uc5d0 \uc218\ub834\ud558\ub294 \uc720\ub9ac\uc218\uc5f4 \\(\\left\\{ r_n \\right\\}\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \\(\\exp\\)\uc640 \\(e^x\\)\uc740 \ubaa8\ub450 \uc5f0\uc18d\ud568\uc218\uc774\ubbc0\ub85c<br \/>\n\\[e^x = \\lim_{n\\to\\infty} e^{r_n} = \\lim_{n\\to\\infty} \\exp\\left( r_n \\right) = \\exp(x)\\]<br \/>\n\uc774\ub2e4. \uc774\ub85c\uc368 \ub2e4\uc74c \ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"box oneline\">\n\\[\\exp(x) = e^x \\,\\,\\, (x\\in\\mathbb{R})\\tag{4}\\]\n<\/div>\n<p>\ub530\ub77c\uc11c \\(\\exp\\)\ub294 \uace0\ub4f1\ud559\uad50 \uacfc\uc815\uc5d0\uc11c \ub2e4\ub8ec \uc9c0\uc218\ud568\uc218 \\(e^x\\)\uc640 \ub3d9\uc77c\ud55c \uc9c0\uc218\ud568\uc218\uc774\ub2e4.<\/p>\n<p><!-- ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"naturallogarithm\"><\/a><\/p>\n<h3>\uc790\uc5f0\ub85c\uadf8\ud568\uc218<\/h3>\n<p>\\(x \\ge 1\\)\uc77c \ub54c<br \/>\n\\[e^x = \\exp(x) = \\sum_{n=0}^{\\infty} \\frac{x^n}{n!} \\ge x\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\lim_{x\\to\\infty} e^x = +\\infty\\tag{5}\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4. \ub610\ud55c<br \/>\n\\[\\lim_{x\\to -\\infty} e^x = \\lim_{x\\to\\infty} e^{-x} = \\lim _ {x\\to\\infty} \\frac{1}{e^x} = 0\\tag{6}\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4. \uadf8\ub7f0\ub370 \\(e^x\\)\uc740 \\(\\mathbb{R}\\)\uc5d0\uc11c \uc99d\uac00\ud558\ub294 \ud568\uc218\uc774\ubbc0\ub85c \\(e^x\\)\uc740 \uc815\uc758\uc5ed\uc774 \\(\\mathbb{R}\\)\uc774\uace0 \uce58\uc5ed\uc774 \\((0,\\,\\infty)\\)\uc778 \uc77c\ub300\uc77c \ud568\uc218\uc774\ub2e4. \ub530\ub77c\uc11c \\(e^x\\)\uc758 \uc5ed\ud568\uc218\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 2. (\uc790\uc5f0\ub85c\uadf8\ud568\uc218)<\/span><\/p>\n<p>\uc784\uc758\uc758 \uc591\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\ln x := \\exp^{-1} (x)\\tag{7}\\]<br \/>\n\ub85c \uc815\uc758\ub41c \ud568\uc218 \\(\\ln\\)\uc744 <span class=\"defined\">\uc790\uc5f0\ub85c\uadf8\ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n<p>\uc790\uc5f0\ub85c\uadf8\ud568\uc218\uc758 \uc131\uc9c8\uc740 \ub300\ubd80\ubd84 \uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \uc131\uc9c8\ub85c\ubd80\ud130 \uc5bb\uc5b4\uc9c4\ub2e4. \\(\\exp\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(\\frac{d}{dx}\\exp (x) > 0\\)\uc774\ubbc0\ub85c \\(\\ln\\)\ub3c4 \uc815\uc758\uc5ed\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70 \\(\\frac{d}{dx} \\ln x > 0\\)\uc774\ub2e4. \uc989 \\(\\ln\\)\uc740 \uc99d\uac00\ud568\uc218\uc774\ub2e4. \ub610\ud55c \\(\\exp\\)\uc758 \uc815\uc758\uc5ed\uc774 \\(\\mathbb{R}\\)\uc774\uace0 \uce58\uc5ed\uc774 \\((0,\\,\\infty )\\)\uc774\ubbc0\ub85c \\(\\ln\\)\uc758 \uc815\uc758\uc5ed\uc740 \\((0,\\,\\infty)\\)\uc774\uace0 \uce58\uc5ed\uc740 \\(\\mathbb{R}\\)\uc774\ub2e4.<\/p>\n<p>\ub610\ud55c \uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \uc131\uc9c8\ub85c\ubd80\ud130 \ub2e4\uc74c \uc131\uc9c8\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 2. (\uc790\uc5f0\ub85c\uadf8\ud568\uc218\uc758 \uc131\uc9c8)<\/span><\/p>\n<p>\ud568\uc218 \\(\\ln\\)\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<ol class=\"bracket\">\n<li>\\(\\ln\\)\uc740 \\((0,\\,\\infty )\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70 \uc99d\uac00\ud558\ub294 \ud568\uc218\uc774\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \uc591\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(\\frac{d}{dx} \\ln x = \\frac{1}{x}\\)\uc774\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \uc591\uc218 \\(x,\\) \\(y\\)\uc5d0 \ub300\ud558\uc5ec \\(\\ln (xy) = \\ln x + \\ln y\\)\uc774\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \uc591\uc218 \\(x\\)\uc640 \uc2e4\uc218 \\(r\\)\uc5d0 \ub300\ud558\uc5ec \\(\\ln \\left( x^r \\right) = r \\ln x\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>[1]\uc740 \ubcf8\ubb38\uc5d0\uc11c \uc99d\uba85\ud558\uc600\ub2e4.<\/p>\n<p>[2] \uc5ed\ud568\uc218 \ubbf8\ubd84\ubc95\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\frac{d}{dx} \\ln x = \\frac{1}{e^{\\ln x}} = \\frac{1}{x}\\]<br \/>\n\uc774\ub2e4.<\/p>\n<p>[3] \uc591\uc218 \\(y\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[\\frac{d}{dx} \\ln(xy) = \\frac{1}{xy} \\cdot y = \\frac{1}{x} = \\frac{d}{dx} \\ln x\\]<br \/>\n\uc774\ubbc0\ub85c \uc0c1\uc218 \\(C\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[\\ln(xy) = \\ln x +C\\]<br \/>\n\uc774\ub2e4. \\(x=1\\)\uc744 \ub300\uc785\ud558\uba74 \\(\\ln y = C\\)\uc774\ubbc0\ub85c<br \/>\n\\[\\ln(xy) = \\ln x + \\ln y\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<p>[4] \uc720\ub9ac\uc218 \\(r\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[\\frac{d}{dx} \\ln \\left( x^r \\right) = \\frac{1}{x^r} \\cdot rx^{r-1} = \\frac{r}{x} = r \\frac{d}{dx} \\ln x \\]<br \/>\n\uc774\ubbc0\ub85c \uc0c1\uc218 \\(C\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[\\ln \\left( x^r \\right) = r \\ln x +C\\]<br \/>\n\uc774\ub2e4. \uc774 \uc2dd\uc758 \uc591\ubcc0\uc5d0 \\(x=1\\)\uc744 \ub300\uc785\ud558\uba74 \\(C=0\\)\uc774\ubbc0\ub85c<br \/>\n\\[\\ln \\left( x^r \\right) = r \\ln x\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<p>\uc774\uc81c \\(r\\)\uac00 \uc2e4\uc218\uc778 \uacbd\uc6b0\ub97c \uc99d\uba85\ud558\uc790. \uc591\uc218 \\(x\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(r\\)\uc5d0 \uc218\ub834\ud558\uace0 \ubaa8\ub4e0 \ud56d\uc774 \uc591\uc218\uc778 \uc720\ub9ac\uc218\uc5f4 \\(\\left\\{ r_n \\right\\}\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \ub85c\uadf8\ud568\uc218\uc640 \uc9c0\uc218\ud568\uc218\ub294 \ubaa8\ub450 \uc5f0\uc18d\ud568\uc218\uc774\ubbc0\ub85c<br \/>\n\\[\\ln\\left(x^r\\right) = \\lim_{n\\to\\infty} \\ln\\left( x^{r_n} \\right) = \\lim_{n\\to\\infty} \\left( r_n \\ln x \\right) = r \\ln x\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"exponentialrule\"><\/a><\/p>\n<h3>\uc9c0\uc218 \ubc95\uce59<\/h3>\n<p>\\(a\\)\uac00 \uc591\uc218\uc774\uace0 \\(x\\)\uac00 \uc2e4\uc218\uc77c \ub54c<br \/>\n\\[a^x = \\exp \\left( \\ln a^x \\right) = \\exp (x \\ln a ) = e^{x \\ln a}\\]<br \/>\n\uc774\ub2e4. \uc989 \uc591\uc218 \\(a\\)\uc640 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<div class=\"box oneline\">\n\\[a^x = e^{x\\ln a} \\,\\,\\, (a > 0 ,\\,\\, x \\in \\mathbb{R} )\\tag{8}\\]\n<\/div>\n<p>\uc9c0\uc218\ud568\uc218\uac00 \uc815\uc758\ub418\uc5b4 \uc788\ub294 \uacbd\uc6b0 (8)\uc740 \uc815\ub9ac\uc774\uc9c0\ub9cc, \uc9c0\uc218\ud568\uc218\uac00 \uc815\uc758\ub418\uc5b4 \uc788\uc9c0 \uc54a\uc740 \uacbd\uc6b0\uc5d0\ub294 (8)\uc744 \uc2e4\uc218 \uc9c0\uc218\uc758 \uc815\uc758\ub85c \uc0bc\uae30\ub3c4 \ud55c\ub2e4. (8)\uc744 \uc774\uc6a9\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\uc774 \uc9c0\uc218\uac00 \uc2e4\uc218\uc778 \uacbd\uc6b0\uc758 \uc9c0\uc218 \ubc95\uce59\uc744 \uc99d\uba85\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 3. (\uc9c0\uc218 \ubc95\uce59)<\/span><\/p>\n<p>\uc591\uc218 \\(a,\\) \\(b\\)\uc640 \uc2e4\uc218 \\(x,\\) \\(y\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"bracket\">\n<li>\\(a^x a^y = a^{x+y}\\)<\/li>\n<li>\\((ab)^x = a^x b^x\\)<\/li>\n<li>\\((a^x )^y = a^{xy}\\)<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\uc815\ub9ac 2\uc640 \ub4f1\uc2dd (8)\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\ln (a^{x+y}) &#038;= (x+y) \\ln a = x \\ln a + y \\ln a \\\\[7pt]<br \/>\n&#038;= \\ln a^x + \\ln a^y = \\ln (a^x a^y ),\\\\[9pt]<br \/>\n\\ln (ab)^x &#038;= x \\ln (ab) = x(\\ln a + \\ln b) \\\\[7pt]<br \/>\n&#038;= x\\ln a + x\\ln b = \\ln a^x + \\ln b^x = \\ln (a^x b^x ), \\\\[9pt]<br \/>\n\\ln (a^x )^y &#038;= y\\ln a^x = xy \\ln a = \\ln a^{xy} .<br \/>\n\\end{align}\\]<br \/>\n\uc5ec\uae30\uc11c \\(\\ln\\)\uc774 \uc77c\ub300\uc77c \ud568\uc218\uc774\ubbc0\ub85c \uc815\ub9ac\uc758 \uc138 \ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"uniqueness\"><\/a><\/p>\n<h3>\uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \uc720\uc77c\uc131<\/h3>\n<p>\uc774\uc81c \uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \uc720\uc77c\uc131\uc744 \uc99d\uba85\ud558\uc790. \ud568\uc218 \\(f : \\mathbb{R} \\to \\mathbb{R}\\)\uac00 \ub2e4\uc74c \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790.<\/p>\n<ul>\n<li>\\(f(0) = 1\\)\uc774\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) > 0\\)\uc774\ub2e4.<\/li>\n<li>\\(f\\)\ub294 \\(\\mathbb{R}\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(f &#8216; (x) = f(x)\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\ub4f1\uc2dd \\(f &#8216; (x) = f(x)\\)\uc758 \uc591\ubcc0\uc744 \\(f(x)\\)\ub85c \ub098\ub204\uba74<br \/>\n\\[\\frac{f &#8216; (x)}{f(x)} = 1\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4. \uc774 \uc2dd\uc758 \uc591\ubcc0\uc758 \ubd80\uc815\uc801\ubd84\uc744 \uad6c\ud558\uba74<br \/>\n\\[\\ln f(x) = x+C\\]<br \/>\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(C\\)\ub294 \uc0c1\uc218\uc774\ub2e4. \uc704 \uc2dd\uc758 \uc591\ubcc0\uc5d0 \\(x=0\\)\uc744 \ub300\uc785\ud558\uba74 \\(C=0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\ln f(x) = x\\]<br \/>\n\uc774\ub2e4. \uc989 \\(f\\)\ub294 \\(\\ln\\)\uc758 \uc5ed\ud568\uc218\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f(x) =e^x\\)\uc774\ub2e4.<\/p>\n<p>\uc774\ub85c\uc368 \ub2e4\uc74c \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 4. (\uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \uc720\uc77c\uc131)<\/span><\/p>\n<p>\ud574\uc11d\uc801\uc73c\ub85c \uc815\uc758\ud55c \uc790\uc5f0\uc9c0\uc218\ud568\uc218\ub294 \uace0\ub4f1\ud559\uad50 \uacfc\uc815\uc5d0\uc11c \uc0b4\ud3b4\ubcf8 \uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc640 \uc644\uc804\ud788 \uc77c\uce58\ud558\uba70, \uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \uc131\uc9c8<br \/>\n\\[f &#8216; (0) = 1 ,\\,\\, f(x) > 0 ,\\,\\, f &#8216; (x) = f(x)\\]<br \/>\n\ub97c \ubaa8\ub450 \uac16\uace0 \uc788\ub294 \ud568\uc218\ub294 (1)\uc5d0\uc11c \uc815\uc758\ud55c \uc790\uc5f0\uc9c0\uc218\ud568\uc218\uac00 \uc720\uc77c\ud558\ub2e4.<\/p>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"generallogarithm\"><\/a><\/p>\n<h3>\uc77c\ubc18\uc801\uc778 \ub85c\uadf8\ud568\uc218<\/h3>\n<p>\uc774\uc81c \uc77c\ubc18\uc801\uc778 \ub85c\uadf8\ud568\uc218\ub97c \uc815\uc758\ud558\uc790.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 3. (\ub85c\uadf8\ud568\uc218)<\/span><\/p>\n<p>\\(a > 0,\\) \\(a\\ne 1\\)\uc77c \ub54c \uc9c0\uc218\ud568\uc218 \\(y=a^x\\)\uc758 \uc5ed\ud568\uc218\ub97c \\(\\log _a x\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc774 \ud568\uc218\ub97c <span class=\"defined\">\ubc11<\/span>\uc774 \\(a\\)\uc778 <span class=\"defined\">\ub85c\uadf8\ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \ud2b9\ud788 \ubc11\uc774 \\(10\\)\uc778 \ub85c\uadf8\ud568\uc218\ub97c <span class=\"defined\">\uc0c1\uc6a9\ub85c\uadf8<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \uc0c1\uc6a9\ub85c\uadf8\ub294 \ubc11\uc744 \uc0dd\ub7b5\ud558\uc5ec \\(\\log_{10} x\\) \ub300\uc2e0 \\(\\log x\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.\n<\/p>\n<\/div>\n<p>\\(a > 0,\\) \\(a \\ne 1,\\) \\(x > 0\\)\uc77c \ub54c, \ub85c\uadf8\ud568\uc218\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(y = \\log_a x\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(x = a^y\\) \uc989<br \/>\n\\[x=e^{y \\ln a}\\]<br \/>\n\uc778 \uac83\uc774\ub2e4. \uc774 \uc2dd\uc758 \uc591\ubcc0\uc5d0 \\(\\ln\\)\uc744 \ucde8\ud558\uba74<br \/>\n\\[\\ln x = \\ln e^{y \\ln a} = y \\ln a\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(y = \\log_a x\\)\uc774\ubbc0\ub85c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\log_a x = \\frac{\\ln x}{\\ln a}\\tag{9}\\]<br \/>\n\uc774 \uc2dd\uc744 \uc774\uc6a9\ud558\uba74 \ubc11\uc774 \\(e\\)\uac00 \uc544\ub2cc \uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud560 \uc218 \uc788\ub2e4. \uc989<br \/>\n\\[<br \/>\n\\frac{d}{dx} a^x = \\frac{d}{dx} e^{x\\ln a} = \\left( e^{x \\ln a} \\right) \\ln a = a^x \\ln a ,\\\\[6pt]<br \/>\n\\frac{d}{dx} \\log_a x = \\frac{d}{dx} \\frac{\\ln x}{\\ln a} = \\frac{1}{x \\ln a}<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/p>\n<p><!--\n\n\n\n\n<div class=\"definition\">\n\n\n<p><span class=\"definition\">\uc815\uc758<\/span><\/p>\n\n\n\n\n<p> ... <\/p>\n\n<\/div>\n\n\n\n\n\n\n\n<div class=\"theorem\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac<\/span><\/p>\n\n\n\n\n<p> ... <\/p>\n\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85<\/p>\n\n\n\n\n<p>\n...\n<span class=\"qed\"><\/span><\/p>\n\n\n<\/div>\n\n\n\n\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 n.<\/span>\n...\n<span class=\"qee\"><\/span>\n<\/p>\n\n\n<\/div>\n\n\n\n\n\n\n<div class=\"remark\">\n\n\n<p><span class=\"remark\">\ucc38\uace0.<\/span>\n...\n<span class=\"qee\"><\/span>\n<\/p>\n\n\n<\/div>\n\n\n\n\n--><\/p>\n<p><!-- . --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc911\ud559\uad50\uc640 \uace0\ub4f1\ud559\uad50 \uacfc\uc815\uc5d0\uc11c\ub294 \uac19\uc740 \uc218\ub97c \uc5ec\ub7ec \ubc88 \uacf1\ud55c \uac83\uc73c\ub85c \uac70\ub4ed\uc81c\uacf1\uc744 \uc815\uc758\ud55c \ub4a4 \uac70\ub4ed\uc81c\uacf1\uc758 \uc9c0\uc218\ub97c \uc815\uc218, \uc720\ub9ac\uc218 \ubc94\uc704\ub85c \ud655\uc7a5\ud558\uba70, \uadf9\ud55c\uc744 \uc774\uc6a9\ud558\uc5ec \uc2e4\uc218 \uc9c0\uc218\ub97c \uc815\uc758\ud55c\ub2e4. \uadf8 \ub4a4\uc5d0 \uc9c0\uc218\ub97c \ubcc0\uc218\ub85c \uac16\ub294 \ud568\uc218\ub97c \uc9c0\uc218\ud568\uc218\ub85c \uc815\uc758\ud558\uba70, \uc9c0\uc218\ud568\uc218\uc758 \uc5ed\ud568\uc218\ub97c \ub85c\uadf8\ud568\uc218\ub85c \uc815\uc758\ud55c\ub2e4. \uc774\uc640 \uac19\uc774 \uc815\uc758\ub41c \uc9c0\uc218\ud568\uc218\ub294 \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uac00 \ub418\uba70, \ud2b9\ud788 \ubc11\uc774 \uc790\uc5f0\uc9c0\uc218\uc778 \ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \\(e^x = 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\frac{x^4}{4!}&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[47],"tags":[353,224,354,359,223,356,355],"class_list":["post-4005","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-353","tag-224","tag-354","tag-359","tag-223","tag-356","tag-355"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4005","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=4005"}],"version-history":[{"count":47,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4005\/revisions"}],"predecessor-version":[{"id":4292,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4005\/revisions\/4292"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=4005"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=4005"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=4005"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}