{"id":1426,"date":"2026-01-28T18:53:40","date_gmt":"2026-01-28T08:53:40","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1426"},"modified":"2026-01-28T18:53:40","modified_gmt":"2026-01-28T08:53:40","slug":"year11-math-3-2-4-exponential-and-logarithmic-functions","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1426","title":{"rendered":"Year11 MATH 3-2-4 Exponential and Logarithmic Functions"},"content":{"rendered":"\n<h4 class=\"wp-block-heading\">Unit2<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Focus: Index laws, log laws, and solving equations.<\/em><\/p>\n\n\n\n<ol start=\"1\" class=\"wp-block-list\">\n<li>Simplify <math data-latex=\"(2^3)^2 \\times 2^{-4}\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>2<\/mn><mn>3<\/mn><\/msup><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>\u00d7<\/mo><msup><mn>2<\/mn><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>4<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">(2^3)^2 \\times 2^{-4}<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li>Solve for <math data-latex=\"x\"><semantics><mi>x<\/mi><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math>: <math data-latex=\"3^{x+1} = 27\"><semantics><mrow><msup><mn>3<\/mn><mrow><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msup><mo>=<\/mo><mn>27<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3^{x+1} = 27<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li>Write <math data-latex=\"\\log_2(16) = 4\"><semantics><mrow><msub><mi>log<\/mi><mn>2<\/mn><\/msub><mo>\u2061<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>16<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\log_2(16) = 4<\/annotation><\/semantics><\/math> in index form.<\/li>\n\n\n\n<li>Solve for <math data-latex=\"x\"><semantics><mi>x<\/mi><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math>: <math data-latex=\"\\log_{10}(x) = 2\"><semantics><mrow><msub><mi>log<\/mi><mn>10<\/mn><\/msub><mo>\u2061<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\log_{10}(x) = 2<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li>Simplify <math data-latex=\"\\log_a(x) + \\log_a(y) - \\log_a(z)\"><semantics><mrow><msub><mi>log<\/mi><mi>a<\/mi><\/msub><mo>\u2061<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>log<\/mi><mi>a<\/mi><\/msub><mo>\u2061<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>y<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><msub><mi>log<\/mi><mi>a<\/mi><\/msub><mo>\u2061<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>z<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\log_a(x) + \\log_a(y) &#8211; \\log_a(z)<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li>Solve <math data-latex=\"e^x = 5\"><semantics><mrow><msup><mi>e<\/mi><mi>x<\/mi><\/msup><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">e^x = 5<\/annotation><\/semantics><\/math> (leave in exact form).<\/li>\n\n\n\n<li>Simplify <math data-latex=\"\\ln(e^3)\"><semantics><mrow><mrow><mi>ln<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mi>e<\/mi><mn>3<\/mn><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\ln(e^3)<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li>Solve <math data-latex=\"2^{2x} - 5(2^x) + 4 = 0\"><semantics><mrow><msup><mn>2<\/mn><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/msup><mo>\u2212<\/mo><mn>5<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>2<\/mn><mi>x<\/mi><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2^{2x} &#8211; 5(2^x) + 4 = 0<\/annotation><\/semantics><\/math> (Hint: let <math data-latex=\"u = 2^x\"><semantics><mrow><mi>u<\/mi><mo>=<\/mo><msup><mn>2<\/mn><mi>x<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">u = 2^x<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n<li>Differentiate <math data-latex=\"y = e^{2x}\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><msup><mi>e<\/mi><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">y = e^{2x}<\/annotation><\/semantics><\/math> (Note: This is usually early Unit 3, but often introduced in Unit 2 &#8220;Methods&#8221; prep).<\/li>\n\n\n\n<li>If <math data-latex=\"\\$1000\"><semantics><mrow><mi>$<\/mi><mn>1000<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\$1000<\/annotation><\/semantics><\/math> is invested at <math data-latex=\"5\\%\"><semantics><mrow><mn>5<\/mn><mi>%<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">5\\%<\/annotation><\/semantics><\/math> p.a. compounding annually, write the formula for the amount <math data-latex=\"A\"><semantics><mi>A<\/mi><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math> after <math data-latex=\"t\"><semantics><mi>t<\/mi><annotation encoding=\"application\/x-tex\">t<\/annotation><\/semantics><\/math> years.<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answers:<\/strong><\/p>\n\n\n\n<ol start=\"1\" class=\"wp-block-list\">\n<li>4 (<math data-latex=\"2^2\"><semantics><msup><mn>2<\/mn><mn>2<\/mn><\/msup><annotation encoding=\"application\/x-tex\">2^2<\/annotation><\/semantics><\/math>) | 2. <math data-latex=\"x = 2\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 2<\/annotation><\/semantics><\/math> | 3. <math data-latex=\"2^4 = 16\"><semantics><mrow><msup><mn>2<\/mn><mn>4<\/mn><\/msup><mo>=<\/mo><mn>16<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2^4 = 16<\/annotation><\/semantics><\/math> | 4. <math data-latex=\"x = 100\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>100<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 100<\/annotation><\/semantics><\/math> | 5. <math data-latex=\"\\log_a(\\frac{xy}{z})\"><semantics><mrow><msub><mi>log<\/mi><mi>a<\/mi><\/msub><mo>\u2061<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mfrac><mrow><mi>x<\/mi><mi>y<\/mi><\/mrow><mi>z<\/mi><\/mfrac><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\log_a(\\frac{xy}{z})<\/annotation><\/semantics><\/math> | 6. <math data-latex=\"x = \\ln(5)\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mrow><mi>ln<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>5<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">x = \\ln(5)<\/annotation><\/semantics><\/math> | 7. 3 | 8. <math data-latex=\"x = 0, x = 2\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><mo separator=\"true\">,<\/mo><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 0, x = 2<\/annotation><\/semantics><\/math> | 9. <math data-latex=\"2e^{2x}\"><semantics><mrow><mn>2<\/mn><msup><mi>e<\/mi><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">2e^{2x}<\/annotation><\/semantics><\/math> | 10. <math data-latex=\"A = 1000(1.05)^t\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mn>1000<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1.05<\/mn><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>t<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">A = 1000(1.05)^t<\/annotation><\/semantics><\/math>.<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Unit2 Focus: Index laws, log laws, and solving equations. Answers:<\/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-1426","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1426","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=1426"}],"version-history":[{"count":4,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1426\/revisions"}],"predecessor-version":[{"id":1443,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1426\/revisions\/1443"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1426"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1426"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1426"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}