{"id":5506,"date":"2025-12-28T12:45:00","date_gmt":"2025-12-28T16:45:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=5506"},"modified":"2026-01-09T12:48:16","modified_gmt":"2026-01-09T16:48:16","slug":"algebra-challenge-12-28-2025","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=5506","title":{"rendered":"Algebra Challenge &#8211; 12\/28\/2025"},"content":{"rendered":"\n<p>Find all real values $x$ such that $4^x+6^x=9^x$.<a href=\"javascript:toggle_visibility('alg-chall-12-28-2025')\">\ud83d\udd11<\/a><\/p>\n<div id=\"alg-chall-12-28-2025\" style=\"display:none\"><\/p>\n\n\n\n<p><strong>Solutions: <\/strong>Dividing $6^x$ on both side of the equation, we have: $$\\Big{(}\\dfrac{2}{3}\\Big{)}^x+1=\\Big{(}\\dfrac{3}{2}\\Big{)}^x$$<\/p>\n\n\n\n<p>Let $y=\\Big{(}\\dfrac{2}{3}\\Big{)}^x$, we have $$y+1=\\dfrac{1}{y}$$ i.e. $$y^2+y-1=0$$<\/p>\n\n\n\n<p>Solving the above equation, we have $$y=\\dfrac{-1\\pm\\sqrt{5}}{2}$$ Because $x$ is real, and $y=\\Big{(}\\dfrac{2}{3}\\Big{)}^x>0$, ignoring the negative $y$ value, we have $$\\Big{(}\\dfrac{2}{3}\\Big{)}^x=\\dfrac{\\sqrt{5}-1}{2}$$ Solving the above equation, we have $$x=\\dfrac{\\text{ln}\\Big{(}\\dfrac{\\sqrt{5}-1}{2}\\Big{)}}{\\text{ln}\\dfrac{2}{3}}=\\boxed{\\dfrac{\\text{ln}(\\sqrt{5}-1)-\\text{ln}(2)}{\\text{ln}(2)-\\text{ln}(3)}}$$<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Find all real values $x$ such that $4^x+6^x=9^x$.\ud83d\udd11 Solutions: Dividing $6^x$ on both side of the equation, we have: $$\\Big{(}\\dfrac{2}{3}\\Big{)}^x+1=\\Big{(}\\dfrac{3}{2}\\Big{)}^x$$ Let $y=\\Big{(}\\dfrac{2}{3}\\Big{)}^x$, we have $$y+1=\\dfrac{1}{y}$$ i.e. $$y^2+y-1=0$$ Solving the above equation, we have $$y=\\dfrac{-1\\pm\\sqrt{5}}{2}$$ Because $x$ is real, and $y=\\Big{(}\\dfrac{2}{3}\\Big{)}^x>0$, &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=5506\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[13],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5506"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5506"}],"version-history":[{"count":9,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5506\/revisions"}],"predecessor-version":[{"id":5515,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5506\/revisions\/5515"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5506"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5506"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5506"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}