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Ayrıca, deprem büyüklüğü hesaplama gibi pratik uygulamalar da videoda yer almaktadır."]},"endTime":2151,"title":"AYT Matematik Dersi: Logaritma Denklemleri ve Eşitsizlikler","beginTime":0}],"fullResult":[{"index":0,"title":"Logaritma Denklemleri ve Eşitsizlikler","list":{"type":"unordered","items":["Kampüs kanalında AYT programı devam ediyor ve şimdi logaritmanın denklem ve eşitsizlikler kısmına geçiliyor.","Logaritmanın tabanı, üslü sayıların tabanıdır ve tabanları aynı olan ifadelerde (taban 1'den farklı ve sıfırdan farklı) kuvvetler de aynı olmalıdır.","Üslü sayılarda üstün üstü çarpılır ve yer değiştirildiğinde kuvvet eksilir."]},"beginTime":7,"endTime":144,"href":"/video/preview/4326855592802268433?parent-reqid=1765145573240972-4770892626314396966-balancer-l7leveler-kubr-yp-sas-262-BAL&text=DocsInBox&t=7&ask_summarization=1"},{"index":1,"title":"Üstel Fonksiyon Denklemleri","list":{"type":"unordered","items":["Üstel fonksiyon denklemlerinde değişken değiştirme yöntemi 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çıkarılmalıdır."]},"beginTime":245,"endTime":463,"href":"/video/preview/4326855592802268433?parent-reqid=1765145573240972-4770892626314396966-balancer-l7leveler-kubr-yp-sas-262-BAL&text=DocsInBox&t=245&ask_summarization=1"},{"index":3,"title":"Logaritma Problemi Çözümü","list":{"type":"unordered","items":["Logaritma a/2 = logaritma b/3 = logaritma c/5 eşitliğinde, a×b×c = 243 olduğunda a değeri bulunuyor.","Logaritma ifadeleri k şeklinde ifade edilerek logaritma a = 2k, logaritma b = 3k ve logaritma c = 5k olarak bulunuyor.","Logaritma a×b×c = 10k = logaritma 355 = 5×logaritma 3 denkleminden logaritma 3 = 2k bulunuyor ve logaritma a = logaritma 3 = 2k olduğundan a = 3 olarak hesaplanıyor."]},"beginTime":465,"endTime":636,"href":"/video/preview/4326855592802268433?parent-reqid=1765145573240972-4770892626314396966-balancer-l7leveler-kubr-yp-sas-262-BAL&text=DocsInBox&t=465&ask_summarization=1"},{"index":4,"title":"Doğru Parçaları Üzerindeki Uzunluk Problemi","list":{"type":"unordered","items":["Doğru parçalarında AC = log₈, CD = 2logₓ, AB = logₓ+2 ve BD = log₂x+1 uzunlukları verilmiş ve x değeri bulunuyor.","AC+CD = AB+BD eşitliği kullanılarak log₈ + logₓ = log₂(x+1) + log₂x denklemi elde ediliyor.","Logaritma özellikleri kullanılarak 8x = 4x+2 denklemi çözülüyor ve x = 1/2 olarak bulunuyor."]},"beginTime":636,"endTime":801,"href":"/video/preview/4326855592802268433?parent-reqid=1765145573240972-4770892626314396966-balancer-l7leveler-kubr-yp-sas-262-BAL&text=DocsInBox&t=636&ask_summarization=1"},{"index":5,"title":"Logaritma Denklemi Çözümü","list":{"type":"unordered","items":["ln(2x³) - ln(x) = 8 denklemini sağlayan x değeri bulunuyor.","Küp kök x ifadesi x^(1/3) olarak yazılıyor ve denklem ln(2x) - 1/3 ln(x) = 8 şeklinde düzenleniyor.","x = m olarak değiştirilerek denklem çözülüyor ve x = e^(-8/3) veya x = e^(1/3) olarak bulunuyor."]},"beginTime":801,"endTime":955,"href":"/video/preview/4326855592802268433?parent-reqid=1765145573240972-4770892626314396966-balancer-l7leveler-kubr-yp-sas-262-BAL&text=DocsInBox&t=801&ask_summarization=1"},{"index":6,"title":"Logaritma ve Trigonometrik Bağıntı","list":{"type":"unordered","items":["m bir gerçek sayı olmak üzere e^(4tanm) = x ve e^(3cotm) = y olduğunda x ile y arasındaki bağıntı bulunuyor.","Logaritma özellikleri kullanılarak ln(x) = 4tanm ve ln(y) = 3cotm denklemleri elde ediliyor.","Tanjant ve kotanjant çarpımı 1 olduğu için ln(x)ln(y) = 12tanmcotm = 12 olarak bulunuyor ve x ile y arasındaki bağıntı ln(x) = 12 olarak elde ediliyor."]},"beginTime":955,"endTime":1038,"href":"/video/preview/4326855592802268433?parent-reqid=1765145573240972-4770892626314396966-balancer-l7leveler-kubr-yp-sas-262-BAL&text=DocsInBox&t=955&ask_summarization=1"},{"index":7,"title":"Üslü Denklem Çözümü","list":{"type":"unordered","items":["x üzeri 9 - 12 tane 3 üzeri x + 27 denklemi sağlayan x değerlerinin çarpımı soruluyor.","9 ifadesi 3 üzeri 2 olarak yazılabilir ve denklem 3 üzeri x üzeri 2 - 12 tane 3 üzeri x + 27 = 0 şeklinde düzenlenir.","Değişken değiştirme yöntemiyle m = 3 üzeri x olarak tanımlanır ve denklem m² - 12m + 27 = 0 şeklinde çözülür."]},"beginTime":1047,"endTime":1201,"href":"/video/preview/4326855592802268433?parent-reqid=1765145573240972-4770892626314396966-balancer-l7leveler-kubr-yp-sas-262-BAL&text=DocsInBox&t=1047&ask_summarization=1"},{"index":8,"title":"Logaritma ve Trigonometri Sorusu","list":{"type":"unordered","items":["logₐ sin x = cos x + logₐ cos x = sin x denklemi verilmiş ve tanjant ile kotanjant toplamı soruluyor.","Logaritma taban büyüsü değiştirilerek m = logₐ sin x olarak tanımlanır ve denklem m + 1 = 2 şeklinde çözülür.","m = 1 bulunur, sin x = cos x olur ve tan x = 1, cot x = 1 olarak hesaplanır, toplamları 2'dir."]},"beginTime":1201,"endTime":1353,"href":"/video/preview/4326855592802268433?parent-reqid=1765145573240972-4770892626314396966-balancer-l7leveler-kubr-yp-sas-262-BAL&text=DocsInBox&t=1201&ask_summarization=1"},{"index":9,"title":"Logaritma ve Üslü Denklem","list":{"type":"unordered","items":["x üzeri log₃ x = 27 × x eşitliğini sağlayan x değerlerinin çarpımı soruluyor.","Logaritma üssü ve üslü sayının tabanı yer değiştirilerek log₃ x = logₓ 27 + logₓ x şeklinde yazılır.","Denklem çözülür ve log₃ x = 1 bulunur, x değerlerinin çarpımı 3'tür."]},"beginTime":1353,"endTime":1544,"href":"/video/preview/4326855592802268433?parent-reqid=1765145573240972-4770892626314396966-balancer-l7leveler-kubr-yp-sas-262-BAL&text=DocsInBox&t=1353&ask_summarization=1"},{"index":10,"title":"Logaritmadaki Eşitsizlikler","list":{"type":"unordered","items":["Logaritmadaki eşitsizliklerde iki temel durum vardır: taban 1'den büyükse düz mantık, taban 0,1 ile 1 arasında ise ters mantık uygulanır.","Taban 1'den büyükse, büyük olan taraftaki üs diğerinden daha büyüktür; taban 0,1 ile 1 arasında ise küçük olan tarafın kuvveti daha büyüktür.","Logaritmanın tanımına aykırı değerler (örneğin log₃(1)) çözüm aralığına dahil edilmemelidir."]},"beginTime":1551,"endTime":1664,"href":"/video/preview/4326855592802268433?parent-reqid=1765145573240972-4770892626314396966-balancer-l7leveler-kubr-yp-sas-262-BAL&text=DocsInBox&t=1551&ask_summarization=1"},{"index":11,"title":"Logaritma Eşitsizliği Örneği","list":{"type":"unordered","items":["Log₉(2+7) \u003c log₃(x+2) eşitsizliğinde, tabanlar aynı ve 1'den büyük olduğu için düz mantık uygulanır.","Eşitsizlik çözüldüğünde x² + 2x + 2 \u003c 0 eşitsizliği elde edilir ve çözüm aralığı (-∞, -3) ∪ (1, ∞) olarak bulunur.","Logaritmanın tanımına dikkat edilmezse, çözüm aralığı geçersiz olabilir; bu örnekte çözüm aralığı sadece (1, ∞) olarak belirlenir."]},"beginTime":1664,"endTime":1863,"href":"/video/preview/4326855592802268433?parent-reqid=1765145573240972-4770892626314396966-balancer-l7leveler-kubr-yp-sas-262-BAL&text=DocsInBox&t=1664&ask_summarization=1"},{"index":12,"title":"Deprem Büyüklüğü Hesaplama","list":{"type":"unordered","items":["Mikron cinsinden ölçülen maksimum genlik d ve depremin Richar ölçeğine göre büyüklüğü r, r = log(d) formülüyle hesaplanır.","Genliği 90 milimetre olan depremin Richar ölçeğine göre büyüklüğü, 90 mikron olarak hesaplanır.","Log₃(90000) = 2, log₁₀(1000) = 1 olduğundan, depremin büyüklüğü 4,94 olarak bulunur."]},"beginTime":1863,"endTime":2020,"href":"/video/preview/4326855592802268433?parent-reqid=1765145573240972-4770892626314396966-balancer-l7leveler-kubr-yp-sas-262-BAL&text=DocsInBox&t=1863&ask_summarization=1"},{"index":13,"title":"Ödüllü Sorular","list":{"type":"unordered","items":["Önceki dersin ödülü sorusunda 3 / log₂(720) + log₃(725) + log₅(720) işleminin sonucu log₂₇₀ olarak hesaplanmıştır.","Bugünün ödüllü sorusunda 3 tabanında log₄ ifadesi, 2 tabanında log(x+2) eşitliğini sağlayan x değeri sorulmaktadır."]},"beginTime":2020,"endTime":2131,"href":"/video/preview/4326855592802268433?parent-reqid=1765145573240972-4770892626314396966-balancer-l7leveler-kubr-yp-sas-262-BAL&text=DocsInBox&t=2020&ask_summarization=1"}],"linkTemplate":"/video/preview/4326855592802268433?parent-reqid=1765145573240972-4770892626314396966-balancer-l7leveler-kubr-yp-sas-262-BAL&text=DocsInBox&t=%%timestamp%%&ask_summarization=1"},"isAdultDoc":false,"relatedParams":{"text":"Logaritma Denklem ve Eşitsizlikler | AYT 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