{"id":6986,"date":"2021-07-23T23:22:19","date_gmt":"2021-07-23T14:22:19","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6986"},"modified":"2022-03-06T19:49:27","modified_gmt":"2022-03-06T10:49:27","slug":"exponential-and-logarithmic-functions","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/exponential-and-logarithmic-functions\/","title":{"rendered":"\uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218"},"content":{"rendered":"<div style=\"display: none; visibility: hidden;\">\n<style type=\"text\/css\">\n\timg.mfp-img { background-color: white; }\n<\/style>\n<p><!--\n\\[\n\\newcommand{\\vecf}{{\\mathbf{f}}}\n\\newcommand{\\vecL}{{\\mathbf{L}}}\n\\newcommand{\\vecR}{{\\mathbb{R}}}\n\\newcommand{\\imI}{\\boldsymbol{i}}\n\\]\n-->\n<\/div>\n<div class=\"box itc_intro\">\n<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 0\uc7a5 7\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; <span class=\"itc_viewcontents\">(<a href=\"..\/\">\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n<p>\\(a\\)\uac00 \ubcf5\uc18c\uc218\uc774\uace0 \\(n\\)\uc774 \uc591\uc758 \uc815\uc218\uc77c \ub54c, \uc9c0\uc218\uac00 \uc790\uc5f0\uc218\uc778 \uac70\ub4ed\uc81c\uacf1\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[a^1 = a ,\\quad a^{n+1} = a^n \\cdot a .\\]<br \/>\n\ub9cc\uc57d \\(a\\)\uac00 \uc74c\uc774 \uc544\ub2cc \ubcf5\uc18c\uc218\ub77c\uba74, \uc9c0\uc218\uac00 \\(0\\)\uc778 \uac70\ub4ed\uc81c\uacf1\uacfc \uc9c0\uc218\uac00 \uc74c\uc758 \uc815\uc218\uc778 \uac70\ub4ed\uc81c\uacf1\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[a^0 = 1 ,\\quad a^{-n} = \\frac{1}{a^n}.\\]<br \/>\n\ub9cc\uc57d \\(a\\)\uac00 \uc591\uc758 \uc2e4\uc218\ub77c\uba74 \\(b^n = a\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc591\uc218 \\(b\\)\uac00 \uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[a^{\\frac{1}{n}}=b \\quad \\Longleftrightarrow \\quad \\left( a=b^n \\,\\,\\text{and}\\,\\, b > 0 \\right) \\]<br \/>\n\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \ub9cc\uc57d \\(a\\)\uac00 \uc591\uc758 \uc2e4\uc218\uc774\uace0 \\(m\\)\uacfc \\(n\\)\uc774 \uc591\uc758 \uc815\uc218\ub77c\uba74, \uc9c0\uc218\uac00 \uc720\ub9ac\uc218\uc778 \uac70\ub4ed\uc81c\uacf1\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[a^{\\frac{m}{n}} = \\left( a^{\\frac{1}{n}}\\right)^m<br \/>\n\\quad\\text{and}\\quad<br \/>\na^{-\\frac{m}{n}} = \\frac{1}{a^{\\frac{m}{n}}} .\\]<br \/>\n\ub9cc\uc57d \\(a\\)\uac00 \\(1\\)\ubcf4\ub2e4 \ud070 \uc2e4\uc218\uc774\uace0 \\(r\\)\uac00 \uc2e4\uc218\ub77c\uba74, \uc9c0\uc218\uac00 \uc2e4\uc218\uc778 \uac70\ub4ed\uc81c\uacf1\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[a^r = \\left( \\text{the least upper bound of the set } \\left\\{ a^q \\,\\vert\\, q < r \\,\\,\\text{and}\\,\\, q\\in\\mathbb{Q} \\right\\} \\right) .\\]\n\ub9cc\uc57d \\(0 < a < 1\\)\uc774\uace0 \\(r\\)\uac00 \uc2e4\uc218\ub77c\uba74\n\\[a^r = \\frac{1}{\\left( \\frac{1}{a} \\right)^r}\\]\n\ub85c \uc815\uc758\ud55c\ub2e4. \ub9cc\uc57d \\(a=1\\)\uc774\uba74 \ub2f9\uc5f0\ud788 \\(a^r = 1\\)\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 0.7.1. (\uc9c0\uc218 \ubc95\uce59)<\/span><\/p>\n<p>\\(a > 0,\\) \\(b > 0\\)\uc774\uace0 \\(r,\\) \\(s\\)\uac00 \uc2e4\uc218\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"bracket\">\n<li>\\( a^r a^s = a^{r+s} \\)<\/li>\n<li>\\( (ab)^r = a^r b^r \\)<\/li>\n<li>\\( \\left( a^r \\right)^s = a^{rs} \\)<\/li>\n<\/ol>\n<\/div>\n<p>\\(a\\)\uac00 \uc591\uc218\uc774\uba74 \uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \uac70\ub4ed\uc81c\uacf1 \\(a^x\\)\uc774 \uc815\uc758\ub41c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[f : \\mathbb{R} \\rightarrow \\mathbb{R} ,\\,\\, f(x) = a^x\\]<br \/>\n\uc740 \uc798 \uc815\uc758\ub41c \ud568\uc218\uc774\ub2e4. \uc774\uc640 \uac19\uc740 \ud568\uc218 \\(f\\)\ub97c <span class=\"defined\">\ubc11\uc774 \\(\\boldsymbol{a}\\)\uc778 \uc9c0\uc218\ud568\uc218<\/span>(exponential function of base \\(a\\))\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \ud568\uc218\ub294 \\(a < 1\\)\uc774\uba74 \uac10\uc18c\ud558\ub294 \ud568\uc218\uc774\uace0, \\(a > 1\\)\uc774\uba74 \uc99d\uac00\ud558\ub294 \ud568\uc218\uc774\ub2e4.<\/p>\n<div style=\"margin-top: 1.5em; margin-bottom: 1.5em;\">\n<img fetchpriority=\"high\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2021\/08\/calculus_00-07_explog01-02.png\" alt=\"\" width=\"517\" height=\"183\" class=\"aligncenter size-full wp-image-7103\" \/>\n<\/div>\n<p>\ub9cc\uc57d \\(a = 1\\)\uc774\uba74 \\(f(x)=a^x\\)\uc73c\ub85c \uc815\uc758\ub41c \ud568\uc218 \\(f\\)\ub294 \uc0c1\uc218\ud568\uc218\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \u201c\\(f(x)=a^x\\)\uc774 \uc9c0\uc218\ud568\uc218\uc774\ub2e4\u201d\ub77c\uace0 \ub9d0\ud560 \ub550 \\(a > 0\\)\uacfc \\(a\\ne 1\\)\uc744 \uac00\uc815\ud55c \uac83\uc774\ub77c \uc57d\uc18d\ud55c\ub2e4.<\/p>\n<p>\\(a > 0\\)\uc774\uace0 \\(a\\ne 1\\)\uc774\uba74, \ud568\uc218 \\(x \\mapsto a^x\\)\uc740 \\(\\mathbb{R}\\)\ub85c\ubd80\ud130 \\(\\mathbb{R}^+\\)\ub85c\uc758 \uc77c\ub300\uc77c \ub300\uc751\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc774 \ud568\uc218\uc758 \uc5ed\ud568\uc218\uac00 \uc874\uc7ac\ud558\uba70, \uadf8 \uc5ed\ud568\uc218\ub294 \\(\\mathbb{R}^+\\)\ub85c\ubd80\ud130 \\(\\mathbb{R}\\)\ub85c\uc758 \ud568\uc218\uc774\ub2e4.<\/p>\n<p>\\(a > 0\\)\uc774\uace0 \\(a\\ne 1\\)\uc77c \ub54c, \\(\\log _a x\\)\ub97c \\(x > 0\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[y = \\log _a x \\quad \\Longleftrightarrow \\quad a^y = x .\\]<br \/>\n\uc774\ub54c \uc591\uc218 \\(x\\)\ub97c \\(\\log_a x\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218\ub97c <span class=\"defined\">\ubc11\uc774 \\(\\boldsymbol{a}\\)\uc778 \ub85c\uadf8\ud568\uc218<\/span>(logarithmic function of base \\(a\\))\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \ud568\uc218\ub294 \\(a < 1\\)\uc774\uba74 \uac10\uc18c\ud558\ub294 \ud568\uc218\uc774\uace0, \\(a > 1\\)\uc774\uba74 \uc99d\uac00\ud558\ub294 \ud568\uc218\uc774\ub2e4.<\/p>\n<div style=\"margin-top: 1.5em; margin-bottom: 1.5em;\">\n<img decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2021\/08\/calculus_00-07_explog03-04.png\" alt=\"\" width=\"390\" height=\"219\" class=\"aligncenter size-full wp-image-7104\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/calculus_00-07_explog03-04.png 2340w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/calculus_00-07_explog03-04-300x168.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/calculus_00-07_explog03-04-1024x574.png 1024w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/calculus_00-07_explog03-04-768x431.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/calculus_00-07_explog03-04-1536x861.png 1536w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/calculus_00-07_explog03-04-2048x1148.png 2048w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/calculus_00-07_explog03-04-1920x1077.png 1920w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/calculus_00-07_explog03-04-1170x656.png 1170w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/08\/calculus_00-07_explog03-04-585x328.png 585w\" sizes=\"(max-width: 390px) 100vw, 390px\" \/>\n<\/div>\n<p>\\(a > 0\\)\uc774\uace0 \\(a\\ne 1\\)\uc77c \ub54c, \uc9c0\uc218\ud568\uc218 \\(y = a^x\\)\uc640 \ub85c\uadf8\ud568\uc218 \\(y = \\log_a x\\)\ub294 \ubaa8\ub450 \uc5f0\uc18d\ud568\uc218\uc774\ub2e4. \uc5f0\uc18d\ud568\uc218\uc758 \uc815\uc758\uc640 \uc131\uc9c8\uc740 4\uc7a5\uc5d0\uc11c \uc0b4\ud3b4\ubcfc \uac83\uc774\ub2e4.<\/p>\n<p>\ub85c\uadf8\ud568\uc218\uac00 \uc9c0\uc218\ud568\uc218\uc758 \uc5ed\ud568\uc218\uc774\ubbc0\ub85c, \ub85c\uadf8\ud568\uc218\uc758 \uc131\uc9c8\uc740 \uc9c0\uc218\ubc95\uce59\uc73c\ub85c\ubd80\ud130 \uc720\ub3c4\ub41c\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 0.7.2. (\ub85c\uadf8 \ubc95\uce59)<\/span><\/p>\n<p>\\(a > 0,\\) \\(a\\ne 1,\\) \\(b > 0,\\) \\(b\\ne 1\\)\uc774\uace0 \\(x,\\) \\(y\\)\uac00 \uc591\uc758 \uc2e4\uc218\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\n\\( [1] \\,\\, \\log_a (xy) = \\log_a x + \\log_a y \\)<br \/>\n\\( [2] \\,\\, \\displaystyle \\log_a \\frac{x}{y} = \\log_a x &#8211; \\log_a y \\)<br \/>\n\\( [3] \\,\\, \\log_a (x^r ) = r \\log_a x \\quad\\quad (r\\in\\mathbb{R})\\)<br \/>\n\\( [4] \\,\\, \\log_x y = \\displaystyle \\frac{\\log_a y}{\\log_a x} \\quad\\quad(x\\ne 1) \\)\n<\/p>\n<\/div>\n<p><!-- ##################################################################### --><br \/>\n<!--\n\n\n<h2 class=\"itc_h2\">\uc81c\ubaa9<\/h2>\n\n\n--><\/p>\n<p><!--\n\n\n\n<div class=\"theorem margintop2\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span>\n\n\n<\/p>\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<\/div>\n\n\n\n\n########\n\n\u201c\u201d\n\u2018\u2019\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span>\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<h2 class=\"itc_h2\"><\/h2>\n\n\n\n--><\/p>\n<p><!--\n\n\n<div style=\"display: none; visibility: hidden;\">\n\\[\n\\newcommand{\\Hom}{{\\operatorname{Hom}}}\n\\newcommand{\\Mat}{{\\operatorname{Mat}}}\n\\newcommand{\\proj}{{\\operatorname{proj}}}\n\\newcommand{\\adj}{{\\operatorname{adj}}}\n\\]\n<\/div>\n\n\n--><\/p>\n<div class=\"box itc_prev_next_box\">\n<ul class=\"itc_ul\">\n<li class=\"itc_li_prev\">\uc55e\uc758 \uae00 : <a href=\"..\/trigonometric-functions\">\uc0bc\uac01\ud568\uc218<\/a><\/li>\n<li class=\"itc_li_next\">\ub2e4\uc74c \uae00 : <a href=\"..\/euclidean-spaces\">\uc720\ud074\ub9ac\ub4dc \uacf5\uac04<\/a><\/li>\n<\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 0\uc7a5 7\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30) \\(a\\)\uac00 \ubcf5\uc18c\uc218\uc774\uace0 \\(n\\)\uc774 \uc591\uc758 \uc815\uc218\uc77c \ub54c, \uc9c0\uc218\uac00 \uc790\uc5f0\uc218\uc778 \uac70\ub4ed\uc81c\uacf1\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4. \\(a^1 = a ,\\, a^{n+1} = a^n \\cdot a .\\) \ub9cc\uc57d \\(a\\)\uac00 \uc74c\uc774 \uc544\ub2cc \ubcf5\uc18c\uc218\ub77c\uba74, \uc9c0\uc218\uac00 \\(0\\)\uc778 \uac70\ub4ed\uc81c\uacf1\uacfc \uc9c0\uc218\uac00 \uc74c\uc758 \uc815\uc218\uc778 \uac70\ub4ed\uc81c\uacf1\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4. \\(a^0 = 1 ,\\, a^{-n} = \\frac{1}{a^n}.\\) \ub9cc\uc57d \\(a\\)\uac00 \uc591\uc758 \uc2e4\uc218\ub77c\uba74 \\(b^n = a\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc591\uc218&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":6620,"menu_order":17,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"class_list":["post-6986","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6986","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"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=6986"}],"version-history":[{"count":27,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6986\/revisions"}],"predecessor-version":[{"id":8381,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6986\/revisions\/8381"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6620"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=6986"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}