Cov khoom ntawm logarithms, los yog amazing - tom ntej no mus ...

Qhov yuav tsum tau rau xam nyob rau hauv tus neeg tam sim ntawd, sai li sai raws li nws muaj peev xwm mus yamntxwv cov khoom nyob ib ncig ntawm nws. Nws yuav lam xav tias lub quantitative ntsuam xyuas logic maj coj mus rau lub "ntxiv-rho" qhov yuav tsum tau rau cov hom ntawm xam. Cov ob kauj ruam yooj yim yog qhov tseem ceeb chiv - tag nrho lwm cov manipulations nrog cov zauv hu ua npaug, division, exponentiation , etc. - ib tug yooj yim "mechanization" ntawm ib co computational algorithms, uas yog raws li nyob rau hauv yooj yim xam - "quav-rho". Txawm nws yog, tiam sis cov creation ntawm algorithms rau xam yog ib tug loj kev kawm tau ntawm kev xav, thiab lawv sau phau ntawv yuav mus ib txhis tawm lawv tus cim nyob rau hauv lub cim xeeb ntawm noob neej.

Rau los sis xya centuries dhau los nyob rau hauv lub teb ntawm maritime navigation thiab astronomy muaj ntau zog qhov yuav tsum tau rau loj nyiaj ntawm suav, uas yog tsis xav tsis thoob, vim nws yog lub npe hu mus rau lub Nrab Hnub nyoog txoj kev loj hlob ntawm navigation thiab astronomy. Nyob rau hauv keeping nrog cov kab lus "thov breeds mov" ob peb daim ntawv thom nug muaj lub tswv yim - los hloov mas zog-intensive lag luam los ntawm txawj ob tus xov tooj ib tug yooj yim tas li ntawd (dually suav hais tias yog lub tswv yim los hloov lub division los ntawm kev rho tawm). Cov ua hauj lwm version ntawm tus tshiab xam system twb teem tawm nyob rau hauv 1614 nyob rau hauv lub chaw ua hauj lwm ntawm Dzhona Nepera nrog ib tug heev kawg li title "hauj lwm ntawm lub amazing rooj ntawm logarithms." Ntawm cov hoob kawm, tus ntxiv kev txhim kho ntawm tus tshiab system mus rau hauv thiab nyob, tab sis qhov yooj yim zog ntawm logarithms raug teem tawm ntau Napier. Lub tswv yim ntawm kev xam system siv logarithms yog hais tias yog ib tug series ntawm cov zauv ntaub ntawv ib tug geometric vam meej, lawv logarithms kuj tsim tau ib cov kev vam meej, tab sis xam. Nyob rau hauv lub xub ntiag ntawm pre-tsim ntxhuav txoj kev tshiab ntawm kev sib hais haum yooj yim zog suav, thiab cov thawj swb txoj cai (1620 xyoo) yog tej zaum thawj thaum ub thiab heev npaum lub tshuab xam zauv - ib tug indispensable engineering cuab tam.

Rau pioneering txoj kev yeej ib txwm nrog potholes. Xub thawj, lub logarithm ntawm lub hauv paus tau raug muab tshem ntse thiab lub xam tseeb yog tsawg, tab sis twb nyob rau hauv 1624 lub ua kom zoo lub rooj nrog ib tug zauv puag raug luam tawm. Cov khoom ntawm logarithms yog muab tau los ntawm tseem txiav txim: logarithm ntawm b - C yog ib tug xov tooj uas, thaum cov neeg kawm ntawv ntawm logarithm puag (tooj A), ua nyob rau hauv ib tug xov tooj ntawm b. Classic cov ntaubntawv povthawj siv kev xaiv zoo li: logA (b) = C - uas nyeem raws li nram no: b logarithm, rau lub hauv paus A, yog tus naj npawb ntawm C. Nyob rau hauv thiaj li yuav ua ib tug txiav txim siv cov tsis heev li qub, logarithmic tooj, koj yuav tsum paub ib tug txheej ntawm cov kev cai, lub npe hu ua "cov khoom logarithms. " Nyob rau hauv hauv paus ntsiab lus, tag nrho cov kev cai muaj ib qho subtext - yuav ua li cas ntxiv, rho tawm thiab hloov logarithms. Tam sim no peb paub yuav ua li cas yuav ua rau nws.

Logarithmic pes tsawg thiab ib tug

1. logA (1) = 0, lub logarithm ntawm tus xov tooj ntawm 1 yog sib npaug zos rau 0 yog vim li cas - ib tug ncaj qha tshwm sim ntawm ib tug xov tooj tsa mus rau zero degree.

2. logA (A) = 1, tib logarithm nrog puag tus xov tooj yog 1 - no kuj zoo paub muaj tseeb rau tej xov tooj ntawm cov thawj hwj chim.

Ntxiv thiab rho tawm ntawm logarithms

3. logA (m) + logA (n) = logA (m * n) - lub sum ntawm logarithms yog lub logarithm ntawm ob peb tus xov tooj ntawm chaw ua hauj lwm.

4. logA (m) - logA (n) = logA (m / n) - qhov txawv ntawm lub logarithms ntawm cov zauv, zoo ib yam li yav dhau los ib tug, yog sib npaug zos rau cov logarithm ntawm tus piv ntawm cov zauv.

5. logA (1 / n) = - logA (n), lub logarithm ntawm lub ntxeev ntawm lub logarithm ntawm no tus xov tooj yog sib npaug zos rau "rho tawm". Nws yog ib qho yooj yim mus pom tias qhov no yog qhov tshwm sim ntawm yav tas los qhia 4 rau m = 1.

Nws yog ib qho yooj yim mus pom hais tias cov kev cai yuav tsum tau 3-5 rau ob tog ntawm lub tib lub cav luj.

Lub exponents nyob rau hauv cov nqe lus logarithmic

6. logA (mn) = n * logA (m), cov logarithm ntawm tus xov tooj ntawm degree n yog sib npaug zos rau cov logarithm ntawm no tus xov tooj, multiplied los ntawm tus exponent n.

7. cav (Ac) (b) = (1 / c) * logA (b), yog nyeem raws li "lub logarithm ntawm b, yog hais tias tus puag muaj rau hauv daim Ac, sib npaug zos rau cov khoom ntawm lub logarithm nrog puag b thiab Ib tug xov tooj ntawm rov qab c».

Formula hloov logarithm puag

8. logA (b) = - logC (b) / logc (A), logarithm ntawm b rau lub hauv paus A nyob rau txoj kev hloov mus rau lub hauv paus C yog xam raws li lub quotient ntawm lub logarithm nrog puag b C thiab C cov logarithm nrog puag tooj sib npaug zos rau lub yav dhau los puag A, wherein nrog cov kos npe rau "rho tawm".

Qhov saum toj no logarithms thiab lawv lub zog pub rau ib tug haum daim ntawv thov mus piv rau lub xam ntawm lub loj numeric arrays, li no kom txhob muaj lub sij hawm ntawm lub suav teb thiab muab tau cov neeg.

Nws tsis yog li ntawd nyob rau hauv science thiab engineering thaj chaw ntawm logarithms yog siv rau ib tug ntau yam sawv cev ntawm lub cev tshwm sim. Piv txwv li, dav paub siv tus txheeb ze qhov tseem ceeb - decibels thaum ntsuas suab siv thiab lub teeb nyob rau hauv physics, lub meej magnitude nyob rau hauv astronomy nyob rau hauv pH nyob rau hauv Science News for KIDS thiab lwm tus neeg.

Miv nyuas siv zug logarithmic le caag yooj yim saib yog tias noj, piv txwv li, thiab mus muab tsib-lej 3 "manually" (nyob rau hauv ib kem), siv cov ntxhuav ntawm logarithms rau ib daim ntawv ntawm daim ntawv thiab tus swb txoj cai. Suffice nws hais tias nyob rau hauv lub caij nyoog kawg no cov ntaub ntawv, lub xam yuav muab rau lub dag lub zog ntawm 10 vib nas this Yuav ua li cas yog feem ntau ceeb yog qhov tseeb hais tias nyob rau hauv lub niaj hnub xam zauv cov suav siv sij hawm, tsis yog tsawg dua.