Logarithm: yuav ua li cas xam?

Lub degree ntawm ib tug neeg tus xov tooj yog ib tug zauv sij hawm coined centuries dhau los. Lub ntsuas thiab algebra raws li ob qho kev xaiv - decimal thiab tej yam ntuj tso logarithms. Lawv xam los ntawm txawv cov qauv, cov kab zauv uas tsis yog sau ntawv, yog ib txwm sib npaug zos rau txhua lwm yam. Qhov no qhia tias yog leejtwg qhia txog cov thaj chaw uas yuav pab tau rau lub peev xwm muaj nuj nqi.

Nta thiab tseem ceeb nta

Thaum no tus taw tes qhov txawv kaum paub xyuam xim zoo. Cov ntau heev thiab nrov ntawm cov no yog cov:

• Radicand cav, muab faib los ntawm tus nqi ntawm lub hauv paus yog ib txwm yog tib yam li cov logarithm √.
• cav cov khoom no yog ib txwm sib npaug zos rau cov sum ntawm cov chaw tsim tshuaj paus.
• Lg = coob qhov twg, los ntawm txawj tus xov tooj ntawm cov uas ua tau rau hauv nws.
• Yog hais tias koj noj tam sim ntawd los ntawm lub cav tawn divider yuav LG private.

Nyob rau hauv tas li ntawd, muaj ib tug kab zauv raws li nyob rau hauv lub ntsiab qhia tias yog leejtwg (xam tau tias yog ib tug tseem ceeb), cov kev hloov mus rau lub tshiab los luj thiab ob peb me qauv.

Muab xam los ntawm cov logarithm - ib tug es tej lub hom phiaj, yog li ntawd, mus rau lub koom ua ke ntawm cov khoom nyob rau hauv cov tshuaj yuav tsum tau mus ua tib zoo thiab tsis tu ncua saib lawv ua thiab taub hau. Peb yuav tsum tsis txhob hnov qab txog lub rooj, uas yuav tsum tau mus tas li saib thiab pom muaj yuav tsum tau coj xwb cov ntaub ntawv.

Hom zauv sij hawm

Lub ntsiab sib txawv ntawm zauv "thaum" nyob rau hauv lub puag (a). Yog hais tias nws muaj ib daim duab ntawm 10, uas yog tus zauv cav. Nyob rau hauv lub rov qab cov ntaub ntawv «ib tug» yog hloov mus rau hauv "y" thiab muaj ib tug transcendental thiab irrational yam ntxwv. Nws tseem yog nqi sau cia hais tias lub sij nqi yog xam los ntawm tshwj xeeb kab zauv, qhov twg pov thawj yuav tshawb xav kawm tshaj high school kawm ntawv.

Zauv logarithms ntawm lub hom lug siv nyob rau hauv lub xam ntawm complex qauv. Ua li tag nrho cov lus los pab txhawb cov lus teb thiab kom meej meej qhia cov txheej txheem kev daws qhov teeb meem. Nyob rau tib lub sij hawm ua ntej lub sij hloov mus rau lub lag luam yuav tsum tau los tsim kom tau ib lub cav nyob rau hauv tus txheej txheem daim ntawv. Nyob rau hauv tas li ntawd, txhua lub khw tsev kawm ntawv cov khoom siv koj yuav nrhiav tau ib tug tshwj xeeb txoj kab nrog scale, pab kom daws tau qhov sib npaug ntawm tej complexity. Logarithm Briggovym hu ua tooj, los yog Euler tooj, tom qab tus kws tshawb fawb uas thawj sab thiab luam tawm tus nqi juxtaposition ntawm ob txiav txim.

Ob hom mis

Tag nrho cov hom thiab ntau yam ntawm koj laij kev pab raws qib lus teb muaj ib tug mob sij hawm cav, muaj ib tug nyias muaj nyias ib lub npe thiab kev kawm zauv ntaus ntawv. Exponential kab zauv ib tug yuav luag pes tsawg daim ntawv ntawm cov logarithmic muab xam, thaum saib los ntawm qhov tseeb ntsiab. Cia li ua ntej embodiment muaj xws li ib tug xov tooj ntawm tshwj xeeb, pab kom sai sai to taub cov kev kawm, thiab lub thib ob xyoo cov cav ntawm zoo tib yam degree. Lub laij siv rau tom kawg mis yuav tsum muaj xws li nce mus nce los nqis.

Qhov sib txawv thiab terminology

Ob leeg muaj lawv tus kheej lub ntsiab index nta txawv cov txhua lwm yam:

• Logarithm. Ib feem tseem ceeb ntawm - cov obligatory xub ntiag ntawm ib lub hauv paus. Standard version nqi yog sib npaug zos rau 10. sau tias ib theem zuj zus - teev x los yog lg x.
• Ntuj. Yog hais tias nws puag yog kos npe rau «e», uas yog ib qhov xam nruj me ntsis zoo tib yam rau kab zauv uas n yog sai tsiv mus rau infinity, kwv yees li luaj li cas ntawm cov sib npaug ntawm 2,72. Official qhab nia, saws ob leeg nyob rau lub tsev kawm ntawv thiab nyob rau hauv qhov ntau complex kev qauv, - ln x.
• Txawv. Dhau li yooj yim logarithms tshwm sim hexadecimal thiab binary hom (puag 16 thiab 2, ntsig txog). Muaj complex version nrog ib lub hauv paus index ntawm 64, tsis nyob hauv systematic tswj ntawm adaptive hom, nrog geometric precision ua muab xam rau ntawm zaum kawg tshwm sim.

Lub terminology muaj xws li cov nram qab no ntau nyob rau hauv algebraic teeb meem:

• tus nqi;
• sib cav;
• puag.

Muab xam los ntawm cav

Muaj peb txoj kev uas sai sai thiab hais ua tag nrho cov tsim nyog suav nrhiav tau ib tug tshwm sim nrog paj lub sij hawm ntawm ib tug yuav tsum muaj txoj cai kev txiav txim siab. Chiv, kwv yees logarithm koj txiav txim (scientific ntaub ntawv ntawm tus xov tooj ntawm degree). Txhua zoo nqi yuav muab los ntawm cov kab zauv, qhov twg nws yog sib npaug zos rau cov mantissa (tus xov tooj ntawm 1 mus rau 9) thiab txawj los ntawm kaum rau n-th degree. Qhov no variant muab xam raws li nyob rau hauv ob zauv tseeb:

• khoom thiab sum cav yeej ib txwm muaj nqi tib yam;
• logarithm coj los ntawm tus xov tooj ib tug mus rau kaum, yuav tsis pub tshaj tus nqi ntawm 1 point.

1. Yog hais tias ib tug yuam kev nyob rau hauv lub xam tshwm sim, nws yog yeej tsis tsawg tshaj li ib tug nyob rau hauv cov kev coj ntawm kev rho tawm.
2. Raug enhanced yog ib tug txiav txim siab hais tias LG puag muaj peb tshwm sim zaum kawg - ib tug tsib-kaum ntawm ib chav tsev. Yog li ntawd, tej zauv tus nqi yog siab tshaj 3 yeej ntxiv rau cov lus teb ib yam khoom.
3. Yuav luag zoo meej tseeb yog tiav yog hais tias tus tes yog ib tug tshwj xeeb cov lus uas yuav muab tau yooj yim siv nyob rau hauv lawv cov kev soj ntsuam kev ua ub no. Nws yuav siv tau los nrhiav seb yog dab tsi yog tus zauv logarithm mus rau kaum feem pua ntawm cov thawj xov tooj.

Lub keeb kwm ntawm ib tug tiag tiag cav

Thib kaum rau xyoo pua keenly muaj qhov yuav tsum tau rau ntau cov ntsiab lus tshaj tau paub science nyob rau ntawm lub sij hawm. Qhov no yog tshwj xeeb tshaj yog muaj tseeb ntawm multi-muaj nuj nqis faib thiab npaug ntawm tus xov tooj nrog zoo sib xws, xws li zauv feem. Qhov kawg ntawm tus thib ob ib nrab ntawm cov era ntawm ntau minds tau tuaj mus rau lub xaus hais tias qhov sib ntxiv ntawm tus xov tooj nyob rau hauv ib lub rooj, uas muab piv rau ob tug progressions: xam thiab duab. Nyob rau hauv cov ntaub ntawv no, tag nrho cov yooj yim suav twb abut qhov tseeb tus nqi. Ib yam li ntawd, zaum tau ua ke thiab sib rho.

Tus thawj mention ntawm lg twb muaj nyob rau hauv 1614. Nws ua ib tug pib xyaum ua mathematician Napier lub xeem. Nws yog ib nqi sau cia hais tias, txawm tus poj popularization ntawm cov kev tshwm sim, nyob rau hauv cov mis yuam kev vim tus ignorance ntawm ib co ntawm cov ntsiab lus lawm uas nyob tom qab. Nws pib nrog ib tug rau-daim duab kos. Ze tshaj rau tus to taub ntawm lub logarithm ntawm lub Bernoulli cov kwv tij nyob, thiab legitimizing debut tshwm sim nyob rau hauv lub xyoo pua Euler. Nws kuj muab ib tug feature nyob rau hauv lub tshav pob ntawm kev kawm ntawv.

Lub keeb kwm ntawm lub complex cav

Lg debut attempts mus rau integrate mus rau hauv lub sab masses tau thaum lub kaj ntug ntawm lub 18th xyoo pua, Bernoulli thiab Leibniz. Tab sis lub sam xeeb ntawm theoretical suav, lawv tsis tau ua. Nyob rau lub sij hawm no, tag nrho cov sib cav tswv yim yog los xij, tab sis lub caij nyoog ntsiab txhais ntawm lub xov tooj ntawm tsis appropriated. Sib tham resumed tom qab, tab sis ntawm Euler thiab d'Alembert. Cov yav tas yog, nyob rau hauv tus, pom zoo nrog rau ntau cov ntawm cov lus tseeb uas muab los ntawm qhov tus tsim ntawm tus nqi, tab sis kuv xav hais tias qhov zoo thiab tsis zoo indicators, As yuav tsum sib npaug zos. Nyob rau hauv nruab nrab ntawm no caug xyoo mis nws twb qhia raws li qhov kawg kev xaiv. Ntxiv mus, Euler derivative twb luam tawm logarithm thiab cov thawj graphics kos.

ntxhuav

Thaj chaw ntawm tus xov tooj qhia hais tias tus multi-nqi xov tooj yuav tsis muab, thiab cov cav mus nrhiav tau lawv thiab muab tso rau los ntawm tshwj xeeb ntxhuav.

Tshwj xeeb tshaj yog tseem ceeb cov no daim duab yog rau astronomers uas yog yuam mus ua hauj lwm nrog ib tug ntau yam ntawm sequences. Nyob rau hauv Soviet lub sij hawm logarithm nrhiav nyob rau hauv Bradis sau, tso tawm nyob rau hauv 1921. Tom qab ntawd, nyob rau hauv 1971, muaj ib tug ntawv ntawm Vega.