In QLD Year 12 Mathematical Methods (Unit 3), logarithmic and exponential functions involve studying<\/p>\n\n\n\n
f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>a<\/mi>x<\/mi><\/msup><\/mrow>f of x equals a to the x-th power<\/annotation><\/semantics><\/math>, e<\/mi>x<\/mi><\/msup>e to the x-th power<\/annotation><\/semantics><\/math>, and l<\/mi>o<\/mi>g<\/mi>b<\/mi><\/msub>(<\/mo>x<\/mi>)<\/mo><\/mrow>l o g sub b of x<\/annotation><\/semantics><\/math>, including their graphs, transformations, and logarithmic laws. Key calculus applications include finding derivatives (e<\/mi>x<\/mi><\/msup>,<\/mo>ln<\/mi>(<\/mo>x<\/mi>)<\/mo><\/mrow>e to the x-th power comma l n x<\/annotation><\/semantics><\/math>), solving growth\/decay problems, and using natural logarithms (ln<\/mi>x<\/mi><\/mrow>l n x<\/annotation><\/semantics><\/math>) to solve equations.\u00a0<\/p>\n\n\n\nKey Topics Covered:<\/strong> <\/p>\n\n\n\n\nExponential Functions:<\/strong> Revision of a<\/mi>x<\/mi><\/msup>a to the x-th power<\/annotation><\/semantics><\/math>\ud835\udc4e\ud835\udc65 and detailed study of the natural exponential function f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>e<\/mi>x<\/mi><\/msup><\/mrow>f of x equals e to the x-th power<\/annotation><\/semantics><\/math>, including its derivatives and applications.<\/li>\n\n\n\nLogarithmic Functions:<\/strong> Definition, laws of logarithms, and the natural logarithm function y<\/mi>=<\/mo>ln<\/mi>(<\/mo>x<\/mi>)<\/mo><\/mrow>y equals l n x<\/annotation><\/semantics><\/math> as the inverse of y<\/mi>=<\/mo>e<\/mi>x<\/mi><\/msup><\/mrow>y equals e to the x-th power<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\nGraphs and Transformations:<\/strong> Sketching e<\/mi>x<\/mi><\/msup>e to the x-th power<\/annotation><\/semantics><\/math> and ln<\/mi>(<\/mo>x<\/mi>)<\/mo><\/mrow>l n x<\/annotation><\/semantics><\/math> graphs, including transformations (shifts, reflections).<\/li>\n\n\n\nCalculus Application:<\/strong> Differentiating exponential and logarithmic functions, finding rates of change, and solving related modelling problems.<\/li>\n\n\n\nSolving Equations:<\/strong> Using logarithms to solve exponential equations and vice versa.<\/li>\n\n\n\nApplications:<\/strong> Modelling growth and decay situations (e.g., population growth, radioactive decay). <\/li>\n<\/ul>\n\n\n\nThis unit establishes the foundational tools for modelling non-linear relationships and understanding rates of change within those functions. <\/p>\n\n\n\n****************************************************************************<\/p>\n\n\n\nQLD 12\u5e74\u751f\u306e\u6570\u5b66\u30e1\u30bd\u30c3\u30c9\uff08\u30e6\u30cb\u30c3\u30c83\uff09\u3067\u306f\u3001\u5bfe\u6570\u95a2\u6570\u3068\u6307\u6570\u95a2\u6570\u306b\u3064\u3044\u3066\u3001(f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>a<\/mi>x<\/mi><\/msup><\/mrow>f(x)=a^{x}<\/annotation><\/semantics><\/math>)\u3001(e<\/mi>x<\/mi><\/msup>e^{x}<\/annotation><\/semantics><\/math>)\u3001(l<\/mi>o<\/mi>g<\/mi>b<\/mi><\/msub>(<\/mo>x<\/mi>)<\/mo><\/mrow>log_{b}(x)<\/annotation><\/semantics><\/math>) \u306b\u3064\u3044\u3066\u3001\u30b0\u30e9\u30d5\u3001\u5909\u63db\u3001\u5bfe\u6570\u6cd5\u5247\u3092\u542b\u3081\u3066\u5b66\u7fd2\u3057\u307e\u3059\u3002\u5fae\u7a4d\u5206\u5b66\u306e\u4e3b\u306a\u5fdc\u7528\u3068\u3057\u3066\u306f\u3001\u5c0e\u95a2\u6570 ((<\/mo>e<\/mi>x<\/mi><\/msup>,<\/mo>ln<\/mi>\u2061<\/mo><\/mrow>(<\/mo>x<\/mi>)<\/mo>)<\/mo><\/mrow>(e^{x},\\ln (x))<\/annotation><\/semantics><\/math>) \u306e\u6c42\u3081\u65b9\u3001\u5897\u52a0\/\u6e1b\u5c11\u554f\u984c\u306e\u89e3\u6cd5\u3001\u81ea\u7136\u5bfe\u6570 (<\/mo>ln<\/mi>\u2061<\/mo><\/mspace><\/mrow>x<\/mi>)<\/mo><\/mrow>(\\ln x)<\/annotation><\/semantics><\/math> \u3092\u7528\u3044\u305f\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5\u306a\u3069\u304c\u6319\u3052\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n\u4e3b\u306a\u30c8\u30d4\u30c3\u30af\uff1a<\/p>\n\n\n\n\u6307\u6570\u95a2\u6570\uff1a(a<\/mi>x<\/mi><\/msup>a^{x}<\/annotation><\/semantics><\/math>) \u306e\u5fa9\u7fd2\u3068\u81ea\u7136\u6307\u6570\u95a2\u6570 (f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>e<\/mi>x<\/mi><\/msup><\/mrow>f(x)=e^{x}<\/annotation><\/semantics><\/math>) \u306e\u8a73\u7d30\u306a\u5b66\u7fd2\u3001\u305d\u306e\u5c0e\u95a2\u6570\u3068\u5fdc\u7528\u3002<\/p>\n\n\n\n\u5bfe\u6570\u95a2\u6570\uff1a\u5b9a\u7fa9\u3001\u5bfe\u6570\u306e\u6cd5\u5247\u3001\u305d\u3057\u3066\u81ea\u7136\u5bfe\u6570\u95a2\u6570 (y<\/mi>=<\/mo>ln<\/mi>\u2061<\/mo><\/mrow>(<\/mo>x<\/mi>)<\/mo><\/mrow>y=\\ln (x)<\/annotation><\/semantics><\/math>) \u3092 (y<\/mi>=<\/mo>e<\/mi>x<\/mi><\/msup><\/mrow>y=e^{x}<\/annotation><\/semantics><\/math>) \u306e\u9006\u95a2\u6570\u3068\u3057\u3066\u7406\u89e3\u3059\u308b\u3002<\/p>\n\n\n\n\u30b0\u30e9\u30d5\u3068\u5909\u63db\uff1a(e<\/mi>x<\/mi><\/msup>e^{x}<\/annotation><\/semantics><\/math>) \u3068 (ln<\/mi>\u2061<\/mo><\/mrow>(<\/mo>x<\/mi>)<\/mo><\/mrow>\\ln (x)<\/annotation><\/semantics><\/math>) \u306e\u30b0\u30e9\u30d5\u3092\u63cf\u304d\u3001\u5909\u63db\uff08\u30b7\u30d5\u30c8\u3001\u93e1\u6620\u5909\u63db\uff09\u3092\u884c\u3046\u3002<\/p>\n\n\n\n\u5fae\u7a4d\u5206\u306e\u5fdc\u7528\uff1a\u6307\u6570\u95a2\u6570\u3068\u5bfe\u6570\u95a2\u6570\u306e\u5fae\u5206\u3001\u5909\u5316\u7387\u306e\u6c42\u3081\u65b9\u3001\u95a2\u9023\u3059\u308b\u30e2\u30c7\u30ea\u30f3\u30b0\u554f\u984c\u306e\u89e3\u6cd5\u3092\u5b66\u3076\u3002<\/p>\n\n\n\n\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5\uff1a\u5bfe\u6570\u3092\u7528\u3044\u3066\u6307\u6570\u65b9\u7a0b\u5f0f\u3092\u89e3\u304d\u3001\u307e\u305f\u305d\u306e\u9006\u3082\u884c\u3046\u3002<\/p>\n\n\n\n\u5fdc\u7528\uff1a\u5897\u52a0\u3068\u6e1b\u5c11\u306e\u72b6\u6cc1\u306e\u30e2\u30c7\u30ea\u30f3\u30b0\uff08\u4f8b\uff1a\u4eba\u53e3\u5897\u52a0\u3001\u653e\u5c04\u6027\u5d29\u58ca\uff09\u3002<\/p>\n\n\n\n\u3053\u306e\u5358\u5143\u3067\u306f\u3001\u975e\u7dda\u5f62\u95a2\u4fc2\u3092\u30e2\u30c7\u30ea\u30f3\u30b0\u3057\u3001\u305d\u308c\u3089\u306e\u95a2\u6570\u306b\u304a\u3051\u308b\u5909\u5316\u7387\u3092\u7406\u89e3\u3059\u308b\u305f\u3081\u306e\u57fa\u672c\u7684\u306a\u30c4\u30fc\u30eb\u3092\u78ba\u7acb\u3059\u308b\u3002<\/p>\n\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"In QLD Year 12 Mathematical Methods (Unit 3), logarithmic and exponential functions involve studying f(x)=axf of x equals a to the x-th power, exe to the x-th power, and logb(x)l o g sub b of x, including their graphs, transformations, and logarithmic laws. Key calculus applications include finding derivatives (ex,ln(x)e to the x-th power comma […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1337","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1337","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1337"}],"version-history":[{"count":3,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1337\/revisions"}],"predecessor-version":[{"id":1376,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1337\/revisions\/1376"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1337"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1337"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1337"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Key Topics Covered:<\/strong> <\/p>\n\n\n\n\nExponential Functions:<\/strong> Revision of a<\/mi>x<\/mi><\/msup>a to the x-th power<\/annotation><\/semantics><\/math>\ud835\udc4e\ud835\udc65 and detailed study of the natural exponential function f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>e<\/mi>x<\/mi><\/msup><\/mrow>f of x equals e to the x-th power<\/annotation><\/semantics><\/math>, including its derivatives and applications.<\/li>\n\n\n\nLogarithmic Functions:<\/strong> Definition, laws of logarithms, and the natural logarithm function y<\/mi>=<\/mo>ln<\/mi>(<\/mo>x<\/mi>)<\/mo><\/mrow>y equals l n x<\/annotation><\/semantics><\/math> as the inverse of y<\/mi>=<\/mo>e<\/mi>x<\/mi><\/msup><\/mrow>y equals e to the x-th power<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\nGraphs and Transformations:<\/strong> Sketching e<\/mi>x<\/mi><\/msup>e to the x-th power<\/annotation><\/semantics><\/math> and ln<\/mi>(<\/mo>x<\/mi>)<\/mo><\/mrow>l n x<\/annotation><\/semantics><\/math> graphs, including transformations (shifts, reflections).<\/li>\n\n\n\nCalculus Application:<\/strong> Differentiating exponential and logarithmic functions, finding rates of change, and solving related modelling problems.<\/li>\n\n\n\nSolving Equations:<\/strong> Using logarithms to solve exponential equations and vice versa.<\/li>\n\n\n\nApplications:<\/strong> Modelling growth and decay situations (e.g., population growth, radioactive decay). <\/li>\n<\/ul>\n\n\n\nThis unit establishes the foundational tools for modelling non-linear relationships and understanding rates of change within those functions. <\/p>\n\n\n\n****************************************************************************<\/p>\n\n\n\nQLD 12\u5e74\u751f\u306e\u6570\u5b66\u30e1\u30bd\u30c3\u30c9\uff08\u30e6\u30cb\u30c3\u30c83\uff09\u3067\u306f\u3001\u5bfe\u6570\u95a2\u6570\u3068\u6307\u6570\u95a2\u6570\u306b\u3064\u3044\u3066\u3001(f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>a<\/mi>x<\/mi><\/msup><\/mrow>f(x)=a^{x}<\/annotation><\/semantics><\/math>)\u3001(e<\/mi>x<\/mi><\/msup>e^{x}<\/annotation><\/semantics><\/math>)\u3001(l<\/mi>o<\/mi>g<\/mi>b<\/mi><\/msub>(<\/mo>x<\/mi>)<\/mo><\/mrow>log_{b}(x)<\/annotation><\/semantics><\/math>) \u306b\u3064\u3044\u3066\u3001\u30b0\u30e9\u30d5\u3001\u5909\u63db\u3001\u5bfe\u6570\u6cd5\u5247\u3092\u542b\u3081\u3066\u5b66\u7fd2\u3057\u307e\u3059\u3002\u5fae\u7a4d\u5206\u5b66\u306e\u4e3b\u306a\u5fdc\u7528\u3068\u3057\u3066\u306f\u3001\u5c0e\u95a2\u6570 ((<\/mo>e<\/mi>x<\/mi><\/msup>,<\/mo>ln<\/mi>\u2061<\/mo><\/mrow>(<\/mo>x<\/mi>)<\/mo>)<\/mo><\/mrow>(e^{x},\\ln (x))<\/annotation><\/semantics><\/math>) \u306e\u6c42\u3081\u65b9\u3001\u5897\u52a0\/\u6e1b\u5c11\u554f\u984c\u306e\u89e3\u6cd5\u3001\u81ea\u7136\u5bfe\u6570 (<\/mo>ln<\/mi>\u2061<\/mo><\/mspace><\/mrow>x<\/mi>)<\/mo><\/mrow>(\\ln x)<\/annotation><\/semantics><\/math> \u3092\u7528\u3044\u305f\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5\u306a\u3069\u304c\u6319\u3052\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n\u4e3b\u306a\u30c8\u30d4\u30c3\u30af\uff1a<\/p>\n\n\n\n\u6307\u6570\u95a2\u6570\uff1a(a<\/mi>x<\/mi><\/msup>a^{x}<\/annotation><\/semantics><\/math>) \u306e\u5fa9\u7fd2\u3068\u81ea\u7136\u6307\u6570\u95a2\u6570 (f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>e<\/mi>x<\/mi><\/msup><\/mrow>f(x)=e^{x}<\/annotation><\/semantics><\/math>) \u306e\u8a73\u7d30\u306a\u5b66\u7fd2\u3001\u305d\u306e\u5c0e\u95a2\u6570\u3068\u5fdc\u7528\u3002<\/p>\n\n\n\n\u5bfe\u6570\u95a2\u6570\uff1a\u5b9a\u7fa9\u3001\u5bfe\u6570\u306e\u6cd5\u5247\u3001\u305d\u3057\u3066\u81ea\u7136\u5bfe\u6570\u95a2\u6570 (y<\/mi>=<\/mo>ln<\/mi>\u2061<\/mo><\/mrow>(<\/mo>x<\/mi>)<\/mo><\/mrow>y=\\ln (x)<\/annotation><\/semantics><\/math>) \u3092 (y<\/mi>=<\/mo>e<\/mi>x<\/mi><\/msup><\/mrow>y=e^{x}<\/annotation><\/semantics><\/math>) \u306e\u9006\u95a2\u6570\u3068\u3057\u3066\u7406\u89e3\u3059\u308b\u3002<\/p>\n\n\n\n\u30b0\u30e9\u30d5\u3068\u5909\u63db\uff1a(e<\/mi>x<\/mi><\/msup>e^{x}<\/annotation><\/semantics><\/math>) \u3068 (ln<\/mi>\u2061<\/mo><\/mrow>(<\/mo>x<\/mi>)<\/mo><\/mrow>\\ln (x)<\/annotation><\/semantics><\/math>) \u306e\u30b0\u30e9\u30d5\u3092\u63cf\u304d\u3001\u5909\u63db\uff08\u30b7\u30d5\u30c8\u3001\u93e1\u6620\u5909\u63db\uff09\u3092\u884c\u3046\u3002<\/p>\n\n\n\n\u5fae\u7a4d\u5206\u306e\u5fdc\u7528\uff1a\u6307\u6570\u95a2\u6570\u3068\u5bfe\u6570\u95a2\u6570\u306e\u5fae\u5206\u3001\u5909\u5316\u7387\u306e\u6c42\u3081\u65b9\u3001\u95a2\u9023\u3059\u308b\u30e2\u30c7\u30ea\u30f3\u30b0\u554f\u984c\u306e\u89e3\u6cd5\u3092\u5b66\u3076\u3002<\/p>\n\n\n\n\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5\uff1a\u5bfe\u6570\u3092\u7528\u3044\u3066\u6307\u6570\u65b9\u7a0b\u5f0f\u3092\u89e3\u304d\u3001\u307e\u305f\u305d\u306e\u9006\u3082\u884c\u3046\u3002<\/p>\n\n\n\n\u5fdc\u7528\uff1a\u5897\u52a0\u3068\u6e1b\u5c11\u306e\u72b6\u6cc1\u306e\u30e2\u30c7\u30ea\u30f3\u30b0\uff08\u4f8b\uff1a\u4eba\u53e3\u5897\u52a0\u3001\u653e\u5c04\u6027\u5d29\u58ca\uff09\u3002<\/p>\n\n\n\n\u3053\u306e\u5358\u5143\u3067\u306f\u3001\u975e\u7dda\u5f62\u95a2\u4fc2\u3092\u30e2\u30c7\u30ea\u30f3\u30b0\u3057\u3001\u305d\u308c\u3089\u306e\u95a2\u6570\u306b\u304a\u3051\u308b\u5909\u5316\u7387\u3092\u7406\u89e3\u3059\u308b\u305f\u3081\u306e\u57fa\u672c\u7684\u306a\u30c4\u30fc\u30eb\u3092\u78ba\u7acb\u3059\u308b\u3002<\/p>\n\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"In QLD Year 12 Mathematical Methods (Unit 3), logarithmic and exponential functions involve studying f(x)=axf of x equals a to the x-th power, exe to the x-th power, and logb(x)l o g sub b of x, including their graphs, transformations, and logarithmic laws. Key calculus applications include finding derivatives (ex,ln(x)e to the x-th power comma […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1337","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1337","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1337"}],"version-history":[{"count":3,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1337\/revisions"}],"predecessor-version":[{"id":1376,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1337\/revisions\/1376"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1337"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1337"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1337"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
This unit establishes the foundational tools for modelling non-linear relationships and understanding rates of change within those functions. <\/p>\n\n\n\n
****************************************************************************<\/p>\n\n\n\n
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