Quiz — 6403584616060_énoncé_Série 16-Top50-Logarithme.pdf

Question 1 sur 10 20:00

[{"id":83755,"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":"ln(1) = 0 car e^0 = 1. Cette propriété découle directement de la définition du logarithme.","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":83756,"question":"L'équation ln(x) = 2 admet-elle une solution ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai, car ln(x) = 2 ⇒ x = e^2 ≈ 7,39, qui est un réel strictement positif.","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":83757,"question":"Quelle est la dérivée de f(x) = ln(x) ?","option_a":"1\/x","option_b":"x","option_c":"ln(x)\/x","option_d":"e^x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(x) est 1\/x, une propriété fondamentale en analyse.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"1\/x\", \"b\": \"x\", \"c\": \"ln(x)\/x\", \"d\": \"e^x\"}}","_debug_options_count":4},{"id":83758,"question":"L'inégalité ln(x) \u003E 0 est-elle équivalente à x \u003E 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai, car ln(x) \u003E 0 ⇨ x \u003E e^0 = 1. La fonction logarithme est 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":83759,"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":"ln(e^3) = 3 car ln et exp sont des fonctions réciproques : ln(e^x) = x.","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":83760,"question":"La fonction ln(x) est-elle définie en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux, ln(x) n'est défini que pour x \u003E 0. En x = 0, la fonction tend vers −∞.","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},{"id":83761,"question":"Quelle propriété permet d'écrire ln(√x) = (1\/2)ln(x) ?","option_a":"ln(a^n) = n·ln(a)","option_b":"ln(ab) = ln(a)+ln(b)","option_c":"ln(a\/b) = ln(a)−ln(b)","option_d":"ln(1) = 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est la propriété ln(a^n) = n·ln(a) avec n = 1\/2, car √x = x^(1\/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\": \"ln(a^n) = n·ln(a)\", \"b\": \"ln(ab) = ln(a)+ln(b)\", \"c\": \"ln(a\/b) =","_debug_options_count":4},{"id":83762,"question":"L'équation ln(x) + ln(2) = ln(5) admet-elle une solution ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai, car ln(x) + ln(2) = ln(2x) = ln(5) ⇒ 2x = 5 ⇒ x = 2,5 \u003E 0.","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":83763,"question":"Quelle est la limite de ln(x) lorsque x tend vers +∞ ?","option_a":"+∞","option_b":"−∞","option_c":"0","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction ln(x) tend vers +∞ lorsque x → +∞, car elle croît sans borne.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"+∞\", \"b\": \"−∞\", \"c\": \"0\", \"d\": \"1\"}}","_debug_options_count":4},{"id":83764,"question":"La fonction f(x) = ln(x) est-elle convexe sur ]0, +∞[ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai, car sa dérivée seconde f''(x) = −1\/x² est négative, ce qui implique la convexité.","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}]