Logarithms: piv txwv thiab kev daws teeb meem

2024 Tus sau: Angel Austin | [email protected]. Kawg hloov kho: 2024-01-02 17:48

Raws li koj paub, thaum muab cov lus qhia nrog lub hwj chim, lawv cov exponents ib txwm ntxiv (a^ba^c=a^{b+ c). Txoj cai lej no tau muab los ntawm Archimedes, thiab tom qab ntawd, nyob rau hauv lub xyoo pua 8th, tus lej ntawm Virasen tau tsim ib lub rooj ntawm cov lej ntsuas. Nws yog lawv uas tau ua haujlwm rau kev tshawb pom ntxiv ntawm logarithms. Piv txwv ntawm kev siv cov haujlwm no tuaj yeem pom yuav luag txhua qhov chaw uas nws yuav tsum tau ua kom yooj yim cumbersome multiplication rau yooj yim ntxiv. Yog tias koj siv sijhawm 10 feeb nyeem cov lus no, peb yuav piav qhia rau koj tias logarithms yog dab tsi thiab yuav ua li cas nrog lawv. Cov lus yooj yim thiab siv tau.}

Definition in lej

Lub logarithm yog ib qho kev qhia ntawm daim ntawv hauv qab no: log_ab=c c" rau hauv uas koj yuav tsum tau tsa lub hauv paus "a" txhawm rau txhawm rau thaum kawg tau tus nqi " b". Cia peb tsom xam lub logarithm siv cov piv txwv, cia peb hais tias muaj cov lus qhia _{2 8. Yuav nrhiav tau cov lus teb li cas? Nws yog ib qho yooj yim heev, koj yuav tsum nrhiav kom tau ib tug degree uas los ntawm 2 mus rau qhov yuav tsum tau degree koj tau txais 8. Tau ua ib co kev xam nyob rau hauv koj lub siab, peb tau txais tus naj npawb 3! Thiab nws muaj tseeb, vim2 tsa los rau lub hwj chim ntawm 3 muab lus teb 8.}

Ntau yam ntawm logarithms

Rau ntau tus menyuam kawm ntawv thiab cov tub ntxhais kawm, cov ncauj lus no zoo li nyuaj thiab nkag siab tsis tau, tab sis qhov tseeb, logarithms tsis txaus ntshai, qhov tseem ceeb yog kom nkag siab lawv lub ntsiab lus dav dav thiab nco ntsoov lawv cov khoom thiab qee cov cai. Muaj peb hom kev hais lus logarithmic:

Natural logarithm ln a, qhov twg lub hauv paus yog tus lej Euler (e=2, 7).
Decimal logarithm lg a, qhov twg lub hauv paus yog tus lej 10.
Logarithm ntawm ib tus lej b rau hauv paus a>1.

Txhua ntawm lawv tau daws raws li tus qauv, suav nrog kev ua kom yooj yim, txo qis thiab txo qis rau ib qho logarithm siv logarithmic theorems. Yuav kom tau txais qhov tseeb qhov tseem ceeb ntawm logarithms, ib tus yuav tsum nco ntsoov lawv cov khoom thiab qhov kev txiav txim ntawm kev daws teeb meem.

Txoj cai thiab qee qhov txwv

Nyob rau hauv lej, muaj ntau ntau txoj cai-txheej txheem uas tau txais raws li ib tug axiom, uas yog, lawv tsis sib tham thiab muaj tseeb. Piv txwv li, nws tsis tuaj yeem faib cov lej los ntawm xoom, thiab nws tseem tsis tuaj yeem coj tus lej txawm tias tsis zoo. Logarithms kuj muaj lawv tus kheej txoj cai, raws li koj tuaj yeem kawm tau yooj yim ua haujlwm txawm tias ntev thiab muaj peev xwm ntawm logarithmic kab lus:

lub hauv paus ntawm "a" yuav tsum yog siab tshaj xoom, thiab tib lub sijhawm tsis sib npaug rau 1, txwv tsis pub cov lus yuav poob nws lub ntsiab lus, vim "1" thiab "0" mus rau ib qho degree yeej ib txwm muaj. sib npaug rau lawv cov nqi;
if a > 0, then ab>0,nws hloov tawm tias "c" kuj yuav tsum siab dua xoom.

Yuav daws logarithms li cas?

Piv txwv, muab txoj haujlwm los nrhiav cov lus teb rau qhov sib npaug 10^x=100. Nws yooj yim heev, koj yuav tsum xaiv lub hwj chim, tsa tus lej kaum, peb tau 100. Qhov no, ntawm chav kawm Zoo, quadratic zog! 10^2=100.

Tam sim no cia peb sawv cev qhov kev qhia no ua tus logarithmic. Peb tau txais log_{10 100=2. Thaum daws cov logarithms, txhua qhov kev ua tau zoo sib xws los nrhiav lub zog uas lub hauv paus ntawm lub logarithm yuav tsum tau nkag mus rau kom tau txais tus lej muab.}

Txhawm rau txiav txim siab tus nqi ntawm qhov tsis paub qhov degree, koj yuav tsum kawm paub ua haujlwm li cas nrog cov ntawv kawm tiav. Nws zoo li no:

logarithms piv txwv thiab kev daws teeb meem

Raws li koj tuaj yeem pom, qee qhov exponents tuaj yeem kwv yees intuitively yog tias koj muaj lub siab xav thiab kev paub txog cov lus sib npaug. Txawm li cas los xij, cov txiaj ntsig loj dua yuav xav tau lub rooj fais fab. Nws tuaj yeem siv tau txawm tias los ntawm cov neeg uas tsis nkag siab txhua yam hauv cov ntsiab lus ntawm kev ua lej nyuaj. Sab laug muaj cov lej (piv a), kab saum toj kawg nkaus ntawm tus lej yog tus nqi ntawm lub zog c, uas tus lej a tau tsa. Ntawm qhov kev sib tshuam, cov hlwb txhais cov txiaj ntsig ntawm cov lej uas yog cov lus teb (ac=b). Cia peb ua, piv txwv li, thawj lub xov tooj nrog tus lej 10 thiab square nws, peb tau txais tus nqi 100, uas tau qhia ntawm qhov kev sib tshuam ntawm peb ob lub hlwb. Txhua yam yooj yim thiab yooj yim heev txawm tias tib neeg tiag tiag yuav nkag siab!

Qhov sib npaug thiab qhov tsis sib xws

Nws hloov tawm tias thaum twgNyob rau hauv tej yam kev mob, tus exponent yog lub logarithm. Yog li ntawd, txhua tus lej lej tuaj yeem sau ua qhov sib npaug ntawm logarithmic. Piv txwv li, 3⁴=81 tuaj yeem sau ua tus logarithm ntawm 81 rau hauv paus 3, uas yog plaub (log₃81=4). Rau qhov tsis zoo degrees, cov kev cai yog tib yam: 2 ^-5=1/32 sau ua logarithm, peb tau txais_{2 (1/32)=-5. Ib qho ntawm feem txaus nyiam tshaj plaws ntawm kev ua lej yog lub ntsiab lus ntawm "logarithms". Peb yuav xav txog cov piv txwv thiab cov kev daws teeb meem ntawm kev sib npaug qis me ntsis, tam sim ntawd tom qab kawm lawv cov khoom. Txog tam sim no, cia peb saib seb qhov tsis sib xws zoo li cas thiab yuav ua li cas thiaj paub qhov txawv ntawm qhov sib npaug.}

Cov lus hauv qab no tau muab: log_{2(x-1) > 3 - nws yog qhov tsis sib xws ntawm logarithmic, txij li qhov tsis paub tus nqi "x" nyob rau hauv lub cim ntawm lub logarithm. Cov lus qhia kuj sib piv ob qhov tseem ceeb: lub hauv paus ob lub logarithm ntawm tus lej xav tau ntau dua tus lej peb.}

Qhov sib txawv tseem ceeb tshaj plaws ntawm qhov sib npaug ntawm qhov sib npaug thiab qhov tsis sib xws yog qhov sib npaug nrog cov logarithms (piv txwv - logarithm₂x=√9) _{imply nyob rau hauv cov lus teb ib los yog ntau tshaj tus lej qhov tseem ceeb, thaum thaum daws qhov tsis sib xws, ob qho tib si ntawm cov txiaj ntsig tau txais txiaj ntsig thiab cov ntsiab lus ntawm qhov kev ua haujlwm no tau txiav txim siab. Raws li qhov tshwm sim, cov lus teb tsis yog ib qho yooj yim ntawm cov lej ntawm tus kheej, ib yam li hauv cov lus teb ntawm kab zauv, tab sis txuas ntxiv los yog cov lej.}

cov khoom ntawm logarithms nrog piv txwv

Basic theorems ntawm logarithms

Thaum daws cov haujlwm tseem ceeb los nrhiav cov txiaj ntsig ntawm lub logarithm, koj yuav tsis paub nws cov khoom. Txawm li cas los xij, thaum nws los txog rau kev sib npaug ntawm logarithmic lossis qhov tsis sib xws, ua ntej ntawm tag nrho cov, nws yog ib qho tsim nyog kom nkag siab meej thiab siv tau rau hauv kev xyaum tag nrho cov yam ntxwv ntawm logarithms. Peb yuav tau paub txog cov piv txwv ntawm qhov sib npaug tom qab, cia peb ua ntej txheeb xyuas txhua yam khoom kom ntxaws ntxiv.

Tus neeg paub tab zoo li no: alogaB=B. Nws tsuas yog siv tau yog tias qhov ntau dua 0, tsis sib npaug rau ib qho, thiab B yog siab dua xoom.

Lub logarithm ntawm cov khoom tuaj yeem sawv cev hauv cov qauv hauv qab no: log_d(s₁s_2)=logd_s1 _{+ log}d_s2. _{Nyob hauv qhov no, qhov yuav tsum tau ua yog: d, s}1 _{thiab s}2 _{> 0; ib ≠1. Koj tuaj yeem muab cov ntaub ntawv pov thawj rau cov qauv no ntawm logarithms, nrog rau cov piv txwv thiab kev daws teeb meem. Let log}a_s1 _=f1 _{and log}a_s 2_=f2_{, then a}f1^=s1_{, a} f2^=s2. _{We get that s}1_s2_=af1^a f2^=af1+f2 ^{(degree properties), thiab ntxiv los ntawm txhais: log}a_(s1 _s2_)=f1_{+ f}2_=log a_{s1 + log}a_s2, _{uas yuav tsum tau ua pov thawj.}

Lub logarithm ntawm cov quotient zoo li no: log_a(s_1/s₂)=log _as₁- log_as_2.

Theorem nyob rau hauv daim ntawv ntawm ib tug formula siv cov nram qab no daim ntawv: log_aq_bn ^{=n/q log}a_b.

Cov qauv no yog hu ua "cov khoom ntawm qib ntawm lub logarithm". Nws zoo ib yam li cov khoom ntawm cov qib zoo tib yam, thiab nws tsis yog qhov xav tsis thoob, vim tias txhua qhov kev ua lej nyob ntawm cov postulates tsis tu ncua. Cia peb saib cov pov thawj.

Let log_ab=t, we get a^t=b. Yog koj tsa ob tog rau lub zog m: a^tn=b^;

but vim a^tn=(a^q)^nt/q=b, li no log_a^qbⁿ =(nt)/t, then log_a^qbⁿ =n/q log_{ab. Theorem proven.}

Piv txwv ntawm cov teeb meem thiab kev tsis sib xws

Cov teeb meem feem ntau ntawm logarithm yog cov piv txwv ntawm kev sib npaug thiab qhov tsis sib xws. Lawv muaj nyob rau hauv yuav luag tag nrho cov phau ntawv teeb meem, thiab kuj muaj nyob rau hauv qhov yuav tsum tau ib feem ntawm kev xeem nyob rau hauv lej. Txhawm rau nkag mus rau hauv tsev kawm qib siab lossis xeem dhau kev xeem hauv lej, koj yuav tsum paub daws cov teeb meem no kom raug.

Hmoov tsis zoo, tsis muaj ib txoj kev npaj lossis lub tswv yim los daws thiab txiav txim siab tsis paub tus nqi ntawm lub logarithm, tab sis qee txoj cai tuaj yeem siv rau txhua qhov kev tsis sib xws ntawm lej lossis kev sib npaug ntawm logarithmic. Ua ntej tshaj plaws, koj yuav tsum paub seb qhov kev qhia tuaj yeem ua kom yooj yim lossis txo qis rau hauv daim ntawv dav dav. Koj tuaj yeem ua kom yooj yim cov lus logarithmic ntev yog tias koj siv lawv cov khoom kom raug. Cia wb mus saib lawv sai sai no.

Thaum daws qhov sib npaug ntawm logarithmic,Nws yog ib qho tsim nyog los txiav txim siab seb hom logarithm peb muaj ua ntej peb: ib qho piv txwv ntawm ib qho kev qhia yuav muaj lub logarithm ntuj lossis tus lej lej.

Ntawm no yog cov piv txwv ntawm cov lej lej: ln100, ln1026. Lawv cov kev daws teeb meem poob mus rau qhov tseeb tias koj yuav tsum tau txiav txim siab txog qib uas lub hauv paus 10 yuav sib npaug rau 100 thiab 1026, feem. Rau cov kev daws teeb meem ntawm ntuj logarithms, ib tug yuav tsum siv logarithmic tus kheej los yog lawv cov khoom. Cia peb saib cov piv txwv ntawm kev daws teeb meem logarithmic ntawm ntau hom.

Yuav ua li cas siv cov qauv logarithm: nrog piv txwv thiab cov kev daws

Yog li, cia peb saib cov piv txwv ntawm kev siv lub ntsiab theorems txog logarithms.

Cov cuab yeej ntawm lub logarithm ntawm cov khoom tuaj yeem siv rau hauv cov haujlwm uas nws tsim nyog los decompose tus nqi loj ntawm tus lej b rau hauv cov yam ntxwv yooj yim. Piv txwv li, log₂4 + log₂128=log₂(4128)=log _{2512. Lo lus teb yog 9.}

log₄8=log_{2² 2}3 ^{=3/2 log}2_{2=1, 5 - raws li koj tuaj yeem pom, los ntawm kev siv cov cuab yeej thib plaub ntawm qib ntawm lub logarithm, peb tau tswj hwm los daws qhov pom ntawm thawj zaug. ib tug complex thiab unsolvable qhia. Txhua yam koj yuav tsum tau ua yog qhov tseem ceeb ntawm lub hauv paus thiab tom qab ntawd coj lub zog tawm ntawm lub cim ntawm lub logarithm.}

Khiav los ntawm kev xeem

Logarithms feem ntau pom muaj nyob rau hauv kev xeem nkag, tshwj xeeb tshaj yog muaj teeb meem logarithmic ntau hauv Unified State Examination (xeev xeem rau tag nrho cov tsev kawm ntawv kawm tiav). Feem ntau cov haujlwm no tsis yog nyob rau hauv ntu A nkaus xwb (feem ntauyooj yim xeem ib feem ntawm qhov kev xeem), tab sis kuj nyob rau hauv ib feem C (qhov nyuaj tshaj plaws thiab voluminous hauj lwm). Qhov kev xeem yuav tsum muaj kev paub tseeb thiab zoo meej ntawm lub ncauj lus "Natural logarithms".

Cov piv txwv thiab cov kev daws teeb meem raug coj los ntawm cov qauv ntawm kev xeem. Cia peb saib yuav daws cov haujlwm li cas.

Given log₂(2x-1)=4. Solution:

rewrite the expression, simplifying me ntsis log_2(2x-1)=22^{, los ntawm lub ntsiab lus ntawm logarithm peb tau txais tias 2x-1=2} 4^{, li no 2x=17; x=8, 5.}

Ua raws li ob peb cov lus qhia, ua raws li koj tuaj yeem daws tau yooj yim txhua qhov sib npaug uas muaj cov kab lus uas nyob hauv qab lub cim ntawm lub logarithm.

Nws yog qhov zoo tshaj los txo tag nrho cov logarithms rau tib lub hauv paus kom cov kev daws tsis tau cumbersome thiab tsis meej pem.
Txhua cov lus nyob rau hauv lub logarithm kos npe yog qhia tias yog qhov zoo, yog li thaum muab cov exponent ntawm qhov kev qhia uas nyob rau hauv lub logarithm kos npe thiab raws li nws lub hauv paus, cov lus nyob rau hauv lub logarithm yuav tsum yog qhov zoo.