WP_Roles Object
(
    [roles] => Array
        (
            [administrator] => Array
                (
                    [name] => Operator
                    [capabilities] => Array
                        (
                            [switch_themes] => 1
                            [edit_themes] => 1
                            [activate_plugins] => 1
                            [edit_plugins] => 1
                            [edit_users] => 1
                            [edit_files] => 1
                            [manage_options] => 1
                            [moderate_comments] => 1
                            [manage_categories] => 1
                            [manage_links] => 1
                            [upload_files] => 1
                            [import] => 1
                            [unfiltered_html] => 1
                            [edit_posts] => 1
                            [edit_others_posts] => 1
                            [edit_published_posts] => 1
                            [publish_posts] => 1
                            [edit_pages] => 1
                            [read] => 1
                            [level_10] => 1
                            [level_9] => 1
                            [level_8] => 1
                            [level_7] => 1
                            [level_6] => 1
                            [level_5] => 1
                            [level_4] => 1
                            [level_3] => 1
                            [level_2] => 1
                            [level_1] => 1
                            [level_0] => 1
                            [edit_others_pages] => 1
                            [edit_published_pages] => 1
                            [publish_pages] => 1
                            [delete_pages] => 1
                            [delete_others_pages] => 1
                            [delete_published_pages] => 1
                            [delete_posts] => 1
                            [delete_others_posts] => 1
                            [delete_published_posts] => 1
                            [delete_private_posts] => 1
                            [edit_private_posts] => 1
                            [read_private_posts] => 1
                            [delete_private_pages] => 1
                            [edit_private_pages] => 1
                            [read_private_pages] => 1
                            [delete_users] => 1
                            [create_users] => 1
                            [unfiltered_upload] => 1
                            [edit_dashboard] => 1
                            [update_plugins] => 1
                            [delete_plugins] => 1
                            [install_plugins] => 1
                            [update_themes] => 1
                            [install_themes] => 1
                            [update_core] => 1
                            [list_users] => 1
                            [remove_users] => 1
                            [promote_users] => 1
                            [edit_theme_options] => 1
                            [delete_themes] => 1
                            [export] => 1
                            [copy_posts] => 1
                            [frm_view_forms] => 1
                            [frm_edit_forms] => 1
                            [frm_delete_forms] => 1
                            [frm_change_settings] => 1
                            [frm_view_entries] => 1
                            [frm_delete_entries] => 1
                            [frm_create_entries] => 1
                            [frm_edit_entries] => 1
                            [frm_view_reports] => 1
                            [frm_edit_applications] => 1
                            [frm_application_dashboard] => 1
                            [aioseo_manage_seo] => 1
                            [aioseo_page_analysis] => 1
                            [aioseo_page_general_settings] => 1
                            [aioseo_page_advanced_settings] => 1
                            [aioseo_page_schema_settings] => 1
                            [aioseo_page_social_settings] => 1
                            [aioseo_dashboard] => 1
                            [aioseo_general_settings] => 1
                            [aioseo_search_appearance_settings] => 1
                            [aioseo_social_networks_settings] => 1
                            [aioseo_sitemap_settings] => 1
                            [aioseo_link_assistant_settings] => 1
                            [aioseo_redirects_manage] => 1
                            [aioseo_page_redirects_manage] => 1
                            [aioseo_redirects_settings] => 1
                            [aioseo_seo_analysis_settings] => 1
                            [aioseo_search_statistics_settings] => 1
                            [aioseo_tools_settings] => 1
                            [aioseo_feature_manager_settings] => 1
                            [aioseo_page_link_assistant_settings] => 1
                            [aioseo_page_redirects_settings] => 1
                            [aioseo_local_seo_settings] => 1
                            [aioseo_page_local_seo_settings] => 1
                            [aioseo_page_writing_assistant_settings] => 1
                            [aioseo_about_us_page] => 1
                            [aioseo_setup_wizard] => 1
                            [aioseo_page_seo_revisions_settings] => 1
                            [aioseo_admin] => 1
                            [aioseo_page_ai_content_settings] => 1
                        )

                )

            [editor] => Array
                (
                    [name] => Admin
                    [capabilities] => Array
                        (
                            [moderate_comments] => 1
                            [manage_categories] => 1
                            [manage_links] => 1
                            [upload_files] => 1
                            [unfiltered_html] => 1
                            [edit_posts] => 1
                            [edit_others_posts] => 1
                            [edit_published_posts] => 1
                            [publish_posts] => 1
                            [edit_pages] => 1
                            [read] => 1
                            [level_7] => 1
                            [level_6] => 1
                            [level_5] => 1
                            [level_4] => 1
                            [level_3] => 1
                            [level_2] => 1
                            [level_1] => 1
                            [level_0] => 1
                            [edit_others_pages] => 1
                            [edit_published_pages] => 1
                            [delete_published_pages] => 1
                            [delete_posts] => 1
                            [delete_others_posts] => 1
                            [delete_published_posts] => 1
                            [delete_private_posts] => 1
                            [edit_private_posts] => 1
                            [read_private_posts] => 1
                            [delete_private_pages] => 1
                            [edit_private_pages] => 1
                            [read_private_pages] => 1
                            [list_users] => 1
                            [edit_users] => 1
                            [create_users] => 1
                            [delete_users] => 1
                            [copy_posts] => 1
                            [create_pages] => 1
                            [publish_pages] => 1
                            [delete_pages] => 1
                            [delete_others_pages] => 1
                            [frm_view_forms] => 1
                            [frm_edit_forms] => 1
                            [frm_delete_forms] => 1
                            [frm_view_entries] => 1
                            [frm_delete_entries] => 1
                            [frm_create_entries] => 1
                            [frm_edit_entries] => 1
                            [frm_view_reports] => 1
                            [frm_change_settings] => 1
                            [frm_edit_applications] => 1
                            [frm_application_dashboard] => 1
                            [aioseo_page_analysis] => 1
                            [aioseo_page_general_settings] => 1
                            [aioseo_page_advanced_settings] => 1
                            [aioseo_page_schema_settings] => 1
                            [aioseo_page_social_settings] => 1
                            [aioseo_general_settings] => 1
                            [aioseo_search_appearance_settings] => 1
                            [aioseo_social_networks_settings] => 1
                            [aioseo_redirects_manage] => 1
                            [aioseo_page_redirects_manage] => 1
                            [aioseo_about_us_page] => 1
                            [aioseo_page_seo_revisions_settings] => 1
                            [aioseo_page_ai_content_settings] => 1
                            [aioseo_page_link_assistant_settings] => 1
                        )

                )

            [user] => Array
                (
                    [name] => User
                    [capabilities] => Array
                        (
                            [delete_others_pages] => 
                            [delete_others_posts] => 1
                            [delete_pages] => 
                            [delete_posts] => 1
                            [delete_private_pages] => 1
                            [delete_private_posts] => 1
                            [delete_published_pages] => 1
                            [delete_published_posts] => 1
                            [edit_others_pages] => 1
                            [edit_others_posts] => 1
                            [edit_pages] => 1
                            [edit_posts] => 1
                            [edit_private_pages] => 1
                            [edit_private_posts] => 1
                            [edit_published_pages] => 1
                            [edit_published_posts] => 1
                            [manage_categories] => 1
                            [manage_links] => 1
                            [moderate_comments] => 1
                            [publish_pages] => 
                            [publish_posts] => 1
                            [read] => 1
                            [read_private_pages] => 1
                            [read_private_posts] => 1
                            [unfiltered_html] => 1
                            [upload_files] => 1
                        )

                )

            [student] => Array
                (
                    [name] => Student
                    [capabilities] => Array
                        (
                            [student] => 1
                        )

                )

            [subscriber] => Array
                (
                    [name] => Subscriber
                    [capabilities] => Array
                        (
                            [level_0] => 1
                        )

                )

            [teacher] => Array
                (
                    [name] => Teacher
                    [capabilities] => Array
                        (
                        )

                )

            [aioseo_manager] => Array
                (
                    [name] => SEO Manager
                    [capabilities] => Array
                        (
                            [edit_others_posts] => 1
                            [edit_others_pages] => 1
                            [edit_pages] => 1
                            [edit_posts] => 1
                            [edit_private_pages] => 1
                            [edit_private_posts] => 1
                            [edit_published_pages] => 1
                            [edit_published_posts] => 1
                            [manage_categories] => 1
                            [read_private_pages] => 1
                            [read_private_posts] => 1
                            [read] => 1
                            [aioseo_dashboard] => 1
                            [aioseo_general_settings] => 1
                            [aioseo_sitemap_settings] => 1
                            [aioseo_link_assistant_settings] => 1
                            [aioseo_redirects_manage] => 1
                            [aioseo_page_redirects_manage] => 1
                            [aioseo_redirects_settings] => 1
                            [aioseo_search_statistics_settings] => 1
                            [aioseo_page_analysis] => 1
                            [aioseo_page_general_settings] => 1
                            [aioseo_page_advanced_settings] => 1
                            [aioseo_page_schema_settings] => 1
                            [aioseo_page_social_settings] => 1
                            [aioseo_page_link_assistant_settings] => 1
                            [aioseo_local_seo_settings] => 1
                            [aioseo_page_local_seo_settings] => 1
                            [aioseo_page_writing_assistant_settings] => 1
                            [aioseo_about_us_page] => 1
                            [aioseo_setup_wizard] => 1
                            [aioseo_page_seo_revisions_settings] => 1
                            [aioseo_page_ai_content_settings] => 1
                        )

                )

            [aioseo_editor] => Array
                (
                    [name] => SEO Editor
                    [capabilities] => Array
                        (
                            [edit_others_posts] => 1
                            [edit_others_pages] => 1
                            [edit_pages] => 1
                            [edit_posts] => 1
                            [edit_private_pages] => 1
                            [edit_private_posts] => 1
                            [edit_published_pages] => 1
                            [edit_published_posts] => 1
                            [manage_categories] => 1
                            [read_private_pages] => 1
                            [read_private_posts] => 1
                            [read] => 1
                            [aioseo_page_redirects_manage] => 1
                            [aioseo_page_analysis] => 1
                            [aioseo_page_general_settings] => 1
                            [aioseo_page_advanced_settings] => 1
                            [aioseo_page_schema_settings] => 1
                            [aioseo_page_social_settings] => 1
                            [aioseo_page_link_assistant_settings] => 1
                            [aioseo_page_local_seo_settings] => 1
                            [aioseo_page_writing_assistant_settings] => 1
                            [aioseo_page_seo_revisions_settings] => 1
                            [aioseo_page_ai_content_settings] => 1
                            [aioseo_about_us_page] => 1
                            [aioseo_link_assistant_settings] => 1
                        )

                )

        )

    [role_objects] => Array
        (
            [administrator] => WP_Role Object
                (
                    [name] => administrator
                    [capabilities] => Array
                        (
                            [switch_themes] => 1
                            [edit_themes] => 1
                            [activate_plugins] => 1
                            [edit_plugins] => 1
                            [edit_users] => 1
                            [edit_files] => 1
                            [manage_options] => 1
                            [moderate_comments] => 1
                            [manage_categories] => 1
                            [manage_links] => 1
                            [upload_files] => 1
                            [import] => 1
                            [unfiltered_html] => 1
                            [edit_posts] => 1
                            [edit_others_posts] => 1
                            [edit_published_posts] => 1
                            [publish_posts] => 1
                            [edit_pages] => 1
                            [read] => 1
                            [level_10] => 1
                            [level_9] => 1
                            [level_8] => 1
                            [level_7] => 1
                            [level_6] => 1
                            [level_5] => 1
                            [level_4] => 1
                            [level_3] => 1
                            [level_2] => 1
                            [level_1] => 1
                            [level_0] => 1
                            [edit_others_pages] => 1
                            [edit_published_pages] => 1
                            [publish_pages] => 1
                            [delete_pages] => 1
                            [delete_others_pages] => 1
                            [delete_published_pages] => 1
                            [delete_posts] => 1
                            [delete_others_posts] => 1
                            [delete_published_posts] => 1
                            [delete_private_posts] => 1
                            [edit_private_posts] => 1
                            [read_private_posts] => 1
                            [delete_private_pages] => 1
                            [edit_private_pages] => 1
                            [read_private_pages] => 1
                            [delete_users] => 1
                            [create_users] => 1
                            [unfiltered_upload] => 1
                            [edit_dashboard] => 1
                            [update_plugins] => 1
                            [delete_plugins] => 1
                            [install_plugins] => 1
                            [update_themes] => 1
                            [install_themes] => 1
                            [update_core] => 1
                            [list_users] => 1
                            [remove_users] => 1
                            [promote_users] => 1
                            [edit_theme_options] => 1
                            [delete_themes] => 1
                            [export] => 1
                            [copy_posts] => 1
                            [frm_view_forms] => 1
                            [frm_edit_forms] => 1
                            [frm_delete_forms] => 1
                            [frm_change_settings] => 1
                            [frm_view_entries] => 1
                            [frm_delete_entries] => 1
                            [frm_create_entries] => 1
                            [frm_edit_entries] => 1
                            [frm_view_reports] => 1
                            [frm_edit_applications] => 1
                            [frm_application_dashboard] => 1
                            [aioseo_manage_seo] => 1
                            [aioseo_page_analysis] => 1
                            [aioseo_page_general_settings] => 1
                            [aioseo_page_advanced_settings] => 1
                            [aioseo_page_schema_settings] => 1
                            [aioseo_page_social_settings] => 1
                            [aioseo_dashboard] => 1
                            [aioseo_general_settings] => 1
                            [aioseo_search_appearance_settings] => 1
                            [aioseo_social_networks_settings] => 1
                            [aioseo_sitemap_settings] => 1
                            [aioseo_link_assistant_settings] => 1
                            [aioseo_redirects_manage] => 1
                            [aioseo_page_redirects_manage] => 1
                            [aioseo_redirects_settings] => 1
                            [aioseo_seo_analysis_settings] => 1
                            [aioseo_search_statistics_settings] => 1
                            [aioseo_tools_settings] => 1
                            [aioseo_feature_manager_settings] => 1
                            [aioseo_page_link_assistant_settings] => 1
                            [aioseo_page_redirects_settings] => 1
                            [aioseo_local_seo_settings] => 1
                            [aioseo_page_local_seo_settings] => 1
                            [aioseo_page_writing_assistant_settings] => 1
                            [aioseo_about_us_page] => 1
                            [aioseo_setup_wizard] => 1
                            [aioseo_page_seo_revisions_settings] => 1
                            [aioseo_admin] => 1
                            [aioseo_page_ai_content_settings] => 1
                        )

                )

            [editor] => WP_Role Object
                (
                    [name] => editor
                    [capabilities] => Array
                        (
                            [moderate_comments] => 1
                            [manage_categories] => 1
                            [manage_links] => 1
                            [upload_files] => 1
                            [unfiltered_html] => 1
                            [edit_posts] => 1
                            [edit_others_posts] => 1
                            [edit_published_posts] => 1
                            [publish_posts] => 1
                            [edit_pages] => 1
                            [read] => 1
                            [level_7] => 1
                            [level_6] => 1
                            [level_5] => 1
                            [level_4] => 1
                            [level_3] => 1
                            [level_2] => 1
                            [level_1] => 1
                            [level_0] => 1
                            [edit_others_pages] => 1
                            [edit_published_pages] => 1
                            [delete_published_pages] => 1
                            [delete_posts] => 1
                            [delete_others_posts] => 1
                            [delete_published_posts] => 1
                            [delete_private_posts] => 1
                            [edit_private_posts] => 1
                            [read_private_posts] => 1
                            [delete_private_pages] => 1
                            [edit_private_pages] => 1
                            [read_private_pages] => 1
                            [list_users] => 1
                            [edit_users] => 1
                            [create_users] => 1
                            [delete_users] => 1
                            [copy_posts] => 1
                            [create_pages] => 1
                            [publish_pages] => 1
                            [delete_pages] => 1
                            [delete_others_pages] => 1
                            [frm_view_forms] => 1
                            [frm_edit_forms] => 1
                            [frm_delete_forms] => 1
                            [frm_view_entries] => 1
                            [frm_delete_entries] => 1
                            [frm_create_entries] => 1
                            [frm_edit_entries] => 1
                            [frm_view_reports] => 1
                            [frm_change_settings] => 1
                            [frm_edit_applications] => 1
                            [frm_application_dashboard] => 1
                            [aioseo_page_analysis] => 1
                            [aioseo_page_general_settings] => 1
                            [aioseo_page_advanced_settings] => 1
                            [aioseo_page_schema_settings] => 1
                            [aioseo_page_social_settings] => 1
                            [aioseo_general_settings] => 1
                            [aioseo_search_appearance_settings] => 1
                            [aioseo_social_networks_settings] => 1
                            [aioseo_redirects_manage] => 1
                            [aioseo_page_redirects_manage] => 1
                            [aioseo_about_us_page] => 1
                            [aioseo_page_seo_revisions_settings] => 1
                            [aioseo_page_ai_content_settings] => 1
                            [aioseo_page_link_assistant_settings] => 1
                        )

                )

            [user] => WP_Role Object
                (
                    [name] => user
                    [capabilities] => Array
                        (
                            [delete_others_pages] => 
                            [delete_others_posts] => 1
                            [delete_pages] => 
                            [delete_posts] => 1
                            [delete_private_pages] => 1
                            [delete_private_posts] => 1
                            [delete_published_pages] => 1
                            [delete_published_posts] => 1
                            [edit_others_pages] => 1
                            [edit_others_posts] => 1
                            [edit_pages] => 1
                            [edit_posts] => 1
                            [edit_private_pages] => 1
                            [edit_private_posts] => 1
                            [edit_published_pages] => 1
                            [edit_published_posts] => 1
                            [manage_categories] => 1
                            [manage_links] => 1
                            [moderate_comments] => 1
                            [publish_pages] => 
                            [publish_posts] => 1
                            [read] => 1
                            [read_private_pages] => 1
                            [read_private_posts] => 1
                            [unfiltered_html] => 1
                            [upload_files] => 1
                        )

                )

            [student] => WP_Role Object
                (
                    [name] => student
                    [capabilities] => Array
                        (
                            [student] => 1
                        )

                )

            [subscriber] => WP_Role Object
                (
                    [name] => subscriber
                    [capabilities] => Array
                        (
                            [level_0] => 1
                        )

                )

            [teacher] => WP_Role Object
                (
                    [name] => teacher
                    [capabilities] => Array
                        (
                        )

                )

            [aioseo_manager] => WP_Role Object
                (
                    [name] => aioseo_manager
                    [capabilities] => Array
                        (
                            [edit_others_posts] => 1
                            [edit_others_pages] => 1
                            [edit_pages] => 1
                            [edit_posts] => 1
                            [edit_private_pages] => 1
                            [edit_private_posts] => 1
                            [edit_published_pages] => 1
                            [edit_published_posts] => 1
                            [manage_categories] => 1
                            [read_private_pages] => 1
                            [read_private_posts] => 1
                            [read] => 1
                            [aioseo_dashboard] => 1
                            [aioseo_general_settings] => 1
                            [aioseo_sitemap_settings] => 1
                            [aioseo_link_assistant_settings] => 1
                            [aioseo_redirects_manage] => 1
                            [aioseo_page_redirects_manage] => 1
                            [aioseo_redirects_settings] => 1
                            [aioseo_search_statistics_settings] => 1
                            [aioseo_page_analysis] => 1
                            [aioseo_page_general_settings] => 1
                            [aioseo_page_advanced_settings] => 1
                            [aioseo_page_schema_settings] => 1
                            [aioseo_page_social_settings] => 1
                            [aioseo_page_link_assistant_settings] => 1
                            [aioseo_local_seo_settings] => 1
                            [aioseo_page_local_seo_settings] => 1
                            [aioseo_page_writing_assistant_settings] => 1
                            [aioseo_about_us_page] => 1
                            [aioseo_setup_wizard] => 1
                            [aioseo_page_seo_revisions_settings] => 1
                            [aioseo_page_ai_content_settings] => 1
                        )

                )

            [aioseo_editor] => WP_Role Object
                (
                    [name] => aioseo_editor
                    [capabilities] => Array
                        (
                            [edit_others_posts] => 1
                            [edit_others_pages] => 1
                            [edit_pages] => 1
                            [edit_posts] => 1
                            [edit_private_pages] => 1
                            [edit_private_posts] => 1
                            [edit_published_pages] => 1
                            [edit_published_posts] => 1
                            [manage_categories] => 1
                            [read_private_pages] => 1
                            [read_private_posts] => 1
                            [read] => 1
                            [aioseo_page_redirects_manage] => 1
                            [aioseo_page_analysis] => 1
                            [aioseo_page_general_settings] => 1
                            [aioseo_page_advanced_settings] => 1
                            [aioseo_page_schema_settings] => 1
                            [aioseo_page_social_settings] => 1
                            [aioseo_page_link_assistant_settings] => 1
                            [aioseo_page_local_seo_settings] => 1
                            [aioseo_page_writing_assistant_settings] => 1
                            [aioseo_page_seo_revisions_settings] => 1
                            [aioseo_page_ai_content_settings] => 1
                            [aioseo_about_us_page] => 1
                            [aioseo_link_assistant_settings] => 1
                        )

                )

        )

    [role_names] => Array
        (
            [administrator] => Operator
            [editor] => Admin
            [user] => User
            [student] => Student
            [subscriber] => Subscriber
            [teacher] => Teacher
            [aioseo_manager] => SEO Manager
            [aioseo_editor] => SEO Editor
        )

    [role_key] => sa__user_roles
    [use_db] => 1
    [site_id:protected] => 1
)

Array
(
    [administrator] => Array
        (
            [name] => Operator
            [capabilities] => Array
                (
                    [switch_themes] => 1
                    [edit_themes] => 1
                    [activate_plugins] => 1
                    [edit_plugins] => 1
                    [edit_users] => 1
                    [edit_files] => 1
                    [manage_options] => 1
                    [moderate_comments] => 1
                    [manage_categories] => 1
                    [manage_links] => 1
                    [upload_files] => 1
                    [import] => 1
                    [unfiltered_html] => 1
                    [edit_posts] => 1
                    [edit_others_posts] => 1
                    [edit_published_posts] => 1
                    [publish_posts] => 1
                    [edit_pages] => 1
                    [read] => 1
                    [level_10] => 1
                    [level_9] => 1
                    [level_8] => 1
                    [level_7] => 1
                    [level_6] => 1
                    [level_5] => 1
                    [level_4] => 1
                    [level_3] => 1
                    [level_2] => 1
                    [level_1] => 1
                    [level_0] => 1
                    [edit_others_pages] => 1
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Question 3. Déterminer l’unique proposition vraie parmi celles qui suivent :
A. [ln(3x)]' = ln(3)
B. ln(2+a) = ln(2a), pour tout a > –2
C. ln(2) × ln(a) = ln(a) + ln(2), pour tout a > –2
D. ln(exp(sin(1))) = sin(1)

Explication
On utilise deux propriétés fondamentales :
• ln(exp(u)) = u pour tout réel u
• exp(ln(x)) = x, pour x > 0
Dans la proposition D : ln(exp(sin(1))) = sin(1)
L’identité est donc toujours vraie. Aucun piège : c’est une simple application de ln(exp(x)) = x.

❌ Pourquoi A est fausse
Dérivons correctement :
La proposition donne [ln(3x)]' = 3 / 3x = 1/x.
La proposition A affirme que cela vaut ln(3), ce qui n’a rien à voir : ce n’est pas une dérivée, ce n’est pas une fonction de x, c’est une constante incorrecte.

❌ Pourquoi B est fausse
Erreur classique : on ne peut PAS transformer une addition en multiplication sous un logarithme.
ln(2 + a) ≠ ln(2a)
La seule règle correcte est : ln(x × y) = ln(x) + ln(y)
Mais jamais : ln(x + y) = ln(x) + ln(y)
Donc B est fausse.

❌ Pourquoi C est fausse
La proposition C affirme : ln(2) × ln(a) = ln(a) + ln(2)
C’est impossible car :
• À gauche → produit de deux logarithmes
• À droite → somme de deux logarithmes
Ces opérations n'ont aucun lien algébrique. Ce serait comme dire : 3 × 4 = 3 + 4

Question 3. Déterminer l’unique proposition vraie parmi celles qui suivent : frac{a}{b} = c

f(x) = x^2

E=mc^2

int\space x \space, \space dx

\pi=\frac{a}{b}

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