اللوغاريتمات والدوال اللوغاريتمية - كتاب الرياضيات - الصف 12 - الفصل 1 - المملكة العربية السعودية

{ "language": "ar", "direction": "rtl", "page_context": { "page_title": "اللوغاريتمات والدوال اللوغاريتمية Logarithms and Logarithmic Functions", "page_type": "lesson_content", "main_topics": [ "اللوغاريتمات", "الدوال اللوغاريتمية", "الدالة العكسية" ], "headers": [ "2-3", "فيما سبق:", "والآن:", "المفردات:", "لماذا؟", "الدوال والعبارات اللوغاريتمية:", "مفهوم أساسي", "اللوغاريتم للأساس b", "التعبير اللفظي:", "الرموز:", "مثال:", "إرشادات للدراسة" ], "has_questions": false, "has_formulas": true, "has_examples": true, "has_visual_elements": true }, "sections": [ { "order": 1, "type": "header", "content_classification": "METADATA", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "2-3", "content": "2-3" }, { "order": 2, "type": "sidebar", "content_classification": "METADATA", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "رابط الدرس الرقمي", "content": "رابط الدرس الرقمي\nwww.ien.edu.sa" }, { "order": 3, "type": "title", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "اللوغاريتمات والدوال اللوغاريتمية Logarithms and Logarithmic Functions", "content": "اللوغاريتمات والدوال اللوغاريتمية\nLogarithms and Logarithmic Functions" }, { "order": 4, "type": "sidebar", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "فيما سبق:", "content": "درست إيجاد الدالة العكسية للدالة. (الدرس 1-7)" }, { "order": 5, "type": "sidebar", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "والآن:", "content": "أجد قيمة عبارات لوغاريتمية.\nأمثل دوال لوغاريتمية بيانيًا." }, { "order": 6, "type": "sidebar", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "المفردات:", "content": "اللوغاريتم\nlogarithm\nالدالة اللوغاريتمية\nlogarithmic function" }, { "order": 7, "type": "header", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "لماذا؟", "content": "يرجح كثير من العلماء أن سبب انقراض سلالة الديناصورات هو النيزك الذي ضرب الأرض. ويستعمل الفلكيون مقياس باليرمو (Palermo) لتصنيف أجسام الفضاء كالنيازك وغيرها اعتمادًا على مدى تأثيرها في كوكب الأرض. ولجعل المقارنة بين هذه الأجسام أكثر سهولة تم تطوير المقياس باستعمال الدوال اللوغاريتمية، إذ يمكن إيجاد قيمة مقياس باليرمو PS لجسم فضائي من خلال الدالة R = 10^PS ، حيث R الخطر النسبي الذي يسببه ذلك الجسم، ويمكن كتابة هذه الدالة بصيغة أخرى تسمى الدالة اللوغاريتمية." }, { "order": 8, "type": "main_content", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "EMPTY", "content": "EMPTY" }, { "order": 9, "type": "header", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "الدوال والعبارات اللوغاريتمية:", "content": "يمكنك تمثيل الدالة العكسية للدالة الأسية 2^x = (f(x بيانيًا من خلال تبديل قيم x و y للأزواج المرتبة التي تمثل الدالة." }, { "order": 10, "type": "main_content", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "EMPTY", "content": "EMPTY" }, { "order": 11, "type": "highlight_box", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "EMPTY", "content": "تقترب قيم x من الصفر\nمع تناقص قيم y" }, { "order": 12, "type": "main_content", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "EMPTY", "content": "يظهر من الجدول والتمثيل البياني أعلاه أن الدالة العكسية للدالة 2^x = y هي x = 2^y . وبصورة عامة، فإن الدالة العكسية للدالة b^x = y هي x = b^y . يسمى المتغير y في المعادلة x = b^y لوغاريتم x للأساس b ، ويكتب عادة على الصورة log_b x = y ، ويقرأ y تساوي لوغاريتم x للأساس b ." }, { "order": 13, "type": "header", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "مفهوم أساسي", "content": "EMPTY" }, { "order": 14, "type": "header", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "اللوغاريتم للأساس b", "content": "EMPTY" }, { "order": 15, "type": "definition", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "التعبير اللفظي:", "content": "إذا كان b, x عددين موجبين، حيث 1 ≠ b ، يرمز للوغاريتم x للأساس b بالرمز log_b x . ويعرف على أنه الأس y الذي يجعل المعادلة x = b^y صحيحة." }, { "order": 16, "type": "definition", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "الرموز:", "content": "افترض أن 1 ≠ b > 0 ، فإن: لكل 0 < x يوجد عدد y بحيث" }, { "order": 17, "type": "formula", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "EMPTY", "content": "log_b x = y إذا وفقط إذا كان b^y = x" }, { "order": 18, "type": "example", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "مثال:", "content": "log_3 27 = y → 3^y = 27" }, { "order": 19, "type": "sidebar", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "إرشادات للدراسة", "content": "تسمى y = log_b x الصورة اللوغاريتمية.\nوتسمى x = b^y الصورة الأسية المكافئة لها." }, { "order": 20, "type": "footer", "content_classification": "METADATA", "question_indicators": { "has_question_words": false, "has_numbering": false, "has_multiple_choice": false, "has_instruction_words": false }, "title": "EMPTY", "content": "وزارة التعليم\nالدرس 3-2 اللوغاريتمات والدوال اللوغاريتمية\n97" } ], "visual_elements": [ { "type": "image", "title": "رابط الدرس الرقمي", "description": "A QR code with the text 'رابط الدرس الرقمي' above it and 'www.ien.edu.sa' below it. It links to the digital lesson.", "educational_context": "Provides access to online resources for the lesson.", "estimated": false }, { "type": "image", "title": "EMPTY", "description": "An artistic rendering or photograph of a meteor streaking across a dark, star-filled night sky, with a fiery tail. The background shows a faint glow, possibly from the horizon or atmospheric effects.", "educational_context": "Illustrates the 'لماذا؟' section, which discusses the Palermo scale for classifying space objects like meteors and their impact on Earth, leading to the concept of logarithms.", "estimated": false }, { "type": "graph", "title": "EMPTY", "description": "A Cartesian coordinate graph showing three lines: y = 2^x (blue curve), y = x (red dashed line), and y = log_2 x (green curve). 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