Quiz — 👉Série n°1 Logarithme népérien.pdf

Question 1 sur 10 20:00

[{"id":138816,"question":"Quelle est la valeur de ln(e^3) ?","option_a":"3","option_b":"e^3","option_c":"0","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, ln(e^x) = x pour tout x réel. Ici, ln(e^3) = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3\", \"b\": \"e^3\", \"c\": \"0\", \"d\": \"1\"}}","_debug_options_count":4},{"id":138817,"question":"La fonction ln(x) est définie pour x \u003E 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le logarithme népérien est uniquement défini pour les nombres strictement positifs.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":138818,"question":"Quelle est la dérivée de la fonction f(x) = ln(2x + 1) ?","option_a":"1\/(2x+1)","option_b":"2\/(2x+1)","option_c":"1\/(x+1)","option_d":"2x\/(2x+1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ln(u(x)) est u'(x)\/u(x). Ici, u(x) = 2x + 1, donc u'(x) = 2. Ainsi, f'(x) = 2\/(2x+1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"1\/(2x+1)\", \"b\": \"2\/(2x+1)\", \"c\": \"1\/(x+1)\", \"d\": \"2x\/(2x+1)\"}}","_debug_options_count":4},{"id":138819,"question":"L'équation ln(x) = -2 admet pour solution x = e^(-2).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, si ln(x) = a, alors x = e^a. Ici, x = e^(-2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":138820,"question":"Quelle est la limite de ln(x) lorsque x tend vers 0+ ?","option_a":"0","option_b":"+∞","option_c":"-∞","option_d":"1","option_e":"","option_f":"","bonne_reponse":"c","explication":"La fonction ln(x) tend vers -∞ lorsque x tend vers 0 par la droite.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"+∞\", \"c\": \"-∞\", \"d\": \"1\"}}","_debug_options_count":4},{"id":138821,"question":"Quelle propriété est correcte pour ln(a^b) ?","option_a":"ln(a) + ln(b)","option_b":"ln(a) - ln(b)","option_c":"b·ln(a)","option_d":"ln(a)\/b","option_e":"","option_f":"","bonne_reponse":"c","explication":"La propriété fondamentale est ln(a^b) = b·ln(a), valable pour a \u003E 0 et b réel.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"ln(a) + ln(b)\", \"b\": \"ln(a) - ln(b)\", \"c\": \"b·ln(a)\", \"d\": \"ln(a","_debug_options_count":4},{"id":138822,"question":"La fonction ln(x) est-elle croissante sur son ensemble de définition ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(x) est 1\/x, qui est positive pour x \u003E 0. La fonction est donc strictement croissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":138823,"question":"Résoudre l'inéquation ln(x) ≤ 1.","option_a":"x ≤ e","option_b":"0 \u003C x ≤ e","option_c":"x ≥ e","option_d":"x \u003C 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'inéquation ln(x) ≤ 1 équivaut à x ≤ e, avec x \u003E 0. Donc 0 \u003C x ≤ e.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"x ≤ e\", \"b\": \"0 \u003C x ≤ e\", \"c\": \"x ≥ e\", \"d\": \"x \u003C 1\"}}","_debug_options_count":4},{"id":138824,"question":"Quelle est la valeur de ln(1) ?","option_a":"0","option_b":"1","option_c":"e","option_d":"-1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, ln(1) = 0 car e^0 = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"e\", \"d\": \"-1\"}}","_debug_options_count":4},{"id":138825,"question":"La fonction ln(x) est-elle définie pour x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le logarithme népérien n'est pas défini en x = 0, car ln(0) n'existe pas dans les réels.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]