{
localUrl: '../page/log_base_1.html',
arbitalUrl: 'https://arbital.com/p/log_base_1',
rawJsonUrl: '../raw/41m.json',
likeableId: '2603',
likeableType: 'page',
myLikeValue: '0',
likeCount: '4',
dislikeCount: '0',
likeScore: '4',
individualLikes: [
'EricBruylant',
'JaimeSevillaMolina',
'NateSoares',
'EricRogstad'
],
pageId: 'log_base_1',
edit: '7',
editSummary: '',
prevEdit: '6',
currentEdit: '7',
wasPublished: 'true',
type: 'wiki',
title: 'Logarithm base 1',
clickbait: 'There is no log base 1.',
textLength: '2323',
alias: 'log_base_1',
externalUrl: '',
sortChildrenBy: 'likes',
hasVote: 'false',
voteType: '',
votesAnonymous: 'false',
editCreatorId: 'NateSoares',
editCreatedAt: '2016-09-15 01:27:41',
pageCreatorId: 'NateSoares',
pageCreatedAt: '2016-06-06 22:50:39',
seeDomainId: '0',
editDomainId: 'AlexeiAndreev',
submitToDomainId: '0',
isAutosave: 'false',
isSnapshot: 'false',
isLiveEdit: 'true',
isMinorEdit: 'false',
indirectTeacher: 'false',
todoCount: '0',
isEditorComment: 'false',
isApprovedComment: 'true',
isResolved: 'false',
snapshotText: '',
anchorContext: '',
anchorText: '',
anchorOffset: '0',
mergedInto: '',
isDeleted: 'false',
viewCount: '79',
text: '[summary: There is no [-3nd] base 1, because no matter how many times you multiply 1 by 1, you get 1. If there _were_ a log base 1, it would send 1 to 0 (because $\\log_b(1)=0$ for every $b$), and it would also send 1 to 1 (because $\\log_b(b)=1$ for every $b$), which demonstrates some of the difficulties with $\\log_1.$ In fact, it would need to send 1 to every number, because $\\log(1 \\cdot 1) = \\log(1) + \\log(1)$ and so on. And it would need to send every $x > 1$ to $\\infty$, and every $0 < x < 1$ to $-\\infty,$ and those aren't numbers, so there's no logarithm base 1.\n\nBut _if there was,_ it would be a [-multifunction] with values in the [extended_reals extended real numbers]. This is actually a perfectly valid way to define $\\log_1,$ though doing so is not necessarily a good idea.]\n\nThere is no [-3nd] base 1, because no matter how many times you multiply 1 by 1, you get 1. If there _were_ a log base 1, it would send 1 to 0 (because $\\log_b(1)=0$ for every $b$), and it would also send 1 to 1 (because $\\log_b(b)=1$ for every $b$), which demonstrates some of the difficulties with $\\log_1.$ In fact, it would need to send 1 to every number, because $\\log(1 \\cdot 1) = \\log(1) + \\log(1)$ and so on. And it would need to send every $x > 1$ to $\\infty$, and every $0 < x < 1$ to $-\\infty,$ and those aren't numbers, so there's no logarithm base 1.\n\nBut if you _really_ want a logarithm base $1$, you can define $\\log_1$ to be a multifunction [3js from] [positive_real_numebrs $\\mathbb R^+$] [3lg to] $\\mathbb R \\cup \\{ \\infty, -\\infty \\}.$ On the input $1$ it outputs $\\mathbb R$. On every input $x > 1$ it outputs $\\{ \\infty \\}$. On every input $0 < x < 1$ it outputs $\\{ -\\infty \\}$. This multifunction can be made to satisfy all the basic [4bz properties of the logarithm], if you interpret $=$ as $\\in$, $1^{\\{\\infty\\}}$ as the [interval_notation interval] $(1, \\infty)$, and $1^{\\{-\\infty\\}}$ as the interval $(0, 1)$. For example, $0 \\in \\log_1(1)$, $1 \\in \\log_1(1)$, and $\\log_1(1) + \\log_1(1) \\in \\log_1(1 \\cdot 1)$. $7 \\in log_1(1^7)$, and $15 \\in 1^{\\log_1(15)}$. This is not necessarily the best idea ever, but hey, the [complex_log final form] of the logarithm was already a multifunction, so whatever. See also [log_is_a_multifunction].\n\nWhile you're thinking about weird logarithms, see also [4c8].',
metaText: '',
isTextLoaded: 'true',
isSubscribedToDiscussion: 'false',
isSubscribedToUser: 'false',
isSubscribedAsMaintainer: 'false',
discussionSubscriberCount: '1',
maintainerCount: '1',
userSubscriberCount: '0',
lastVisit: '',
hasDraft: 'false',
votes: [],
voteSummary: 'null',
muVoteSummary: '0',
voteScaling: '0',
currentUserVote: '-2',
voteCount: '0',
lockedVoteType: '',
maxEditEver: '0',
redLinkCount: '0',
lockedBy: '',
lockedUntil: '',
nextPageId: '',
prevPageId: '',
usedAsMastery: 'false',
proposalEditNum: '0',
permissions: {
edit: {
has: 'false',
reason: 'You don't have domain permission to edit this page'
},
proposeEdit: {
has: 'true',
reason: ''
},
delete: {
has: 'false',
reason: 'You don't have domain permission to delete this page'
},
comment: {
has: 'false',
reason: 'You can't comment in this domain because you are not a member'
},
proposeComment: {
has: 'true',
reason: ''
}
},
summaries: {},
creatorIds: [
'NateSoares'
],
childIds: [],
parentIds: [
'logarithm'
],
commentIds: [],
questionIds: [],
tagIds: [
'start_meta_tag'
],
relatedIds: [],
markIds: [],
explanations: [],
learnMore: [],
requirements: [],
subjects: [],
lenses: [],
lensParentId: '',
pathPages: [],
learnMoreTaughtMap: {},
learnMoreCoveredMap: {},
learnMoreRequiredMap: {},
editHistory: {},
domainSubmissions: {},
answers: [],
answerCount: '0',
commentCount: '0',
newCommentCount: '0',
linkedMarkCount: '0',
changeLogs: [
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '19609',
pageId: 'log_base_1',
userId: 'NateSoares',
edit: '7',
type: 'newEdit',
createdAt: '2016-09-15 01:27:41',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15059',
pageId: 'log_base_1',
userId: 'NateSoares',
edit: '6',
type: 'newEdit',
createdAt: '2016-07-01 16:06:33',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15055',
pageId: 'log_base_1',
userId: 'NateSoares',
edit: '5',
type: 'newEdit',
createdAt: '2016-07-01 15:49:37',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14938',
pageId: 'log_base_1',
userId: 'NateSoares',
edit: '4',
type: 'newEdit',
createdAt: '2016-06-30 06:00:15',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14549',
pageId: 'log_base_1',
userId: 'NateSoares',
edit: '3',
type: 'newEdit',
createdAt: '2016-06-24 04:35:39',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '12898',
pageId: 'log_base_1',
userId: 'NateSoares',
edit: '2',
type: 'newEdit',
createdAt: '2016-06-15 05:45:48',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '11853',
pageId: 'log_base_1',
userId: 'NateSoares',
edit: '1',
type: 'newEdit',
createdAt: '2016-06-06 22:50:39',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '11850',
pageId: 'log_base_1',
userId: 'NateSoares',
edit: '1',
type: 'newTag',
createdAt: '2016-06-06 22:50:37',
auxPageId: 'start_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '11849',
pageId: 'log_base_1',
userId: 'NateSoares',
edit: '0',
type: 'deleteTag',
createdAt: '2016-06-06 22:50:33',
auxPageId: 'stub_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '11848',
pageId: 'log_base_1',
userId: 'NateSoares',
edit: '1',
type: 'newTag',
createdAt: '2016-06-06 22:50:03',
auxPageId: 'stub_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '11847',
pageId: 'log_base_1',
userId: 'NateSoares',
edit: '1',
type: 'newParent',
createdAt: '2016-06-06 22:49:59',
auxPageId: 'logarithm',
oldSettingsValue: '',
newSettingsValue: ''
}
],
feedSubmissions: [],
searchStrings: {},
hasChildren: 'false',
hasParents: 'true',
redAliases: {},
improvementTagIds: [],
nonMetaTagIds: [],
todos: [],
slowDownMap: 'null',
speedUpMap: 'null',
arcPageIds: 'null',
contentRequests: {}
}