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Fvnctio logarithmvs eivsqve proprietates

Svmmarivm

Haec lectio logarithmvm basis aa definit vt fvnctionem inversam exponentialis basis aa, et cvm eo logarithmvm natvralem et logarithmvm decimalem. Leges logarithmorvm (prodvcti, qvotientis et potentiae) et formvlam mvtationis basis cvm corollariis eivs demonstrat, monotoniam, signvm et delineationem fvnctionis logarithmi cvm asymptoto verticali investigat, et ostendit log⁡23\log_2 3, et generatim logarithmvm nvmeri natvralis vno maioris in basi natvrali ad evm prima, irrationalem esse. Deinde logarithmvm decimalem adhibet ad digitos nvmeri nvmerandos sine eo scribendo, scalas logarithmicas acvsticae, chemiae et seismologiae describit, et principivm regvlae calcvlatoriae explicat, qvae nvmeros mvltiplicat longitvdines addendo.

Proposita discendi

  1. Logarithmvm vt fvnctionem inversam exponentialis definire, logarithmos simplices compvtare, et dominivm, copiam imaginvm et delineationem fvnctionis logarithmi agnoscere.
  2. Leges logarithmorvm et formvlam mvtationis basis demonstrare et adhibere.
  3. Monotoniam et signvm logarithmi secvndvm basim eivs determinare, et logarithmos per inaeqvalitates inter potentias integras circvmscribere.
  4. Logarithmo decimali vti ad digitos nvmerandos et ad scalas logarithmicas interpretandas, et irrationalitatem qvorvndam logarithmorvm demonstrare.

Logarithmvs vt fvnctio inversa exponentialis

In lectione de fvnctione exponentiali et potentia exponentis realis demonstratvm est, si a>0a > 0 et a≠1a \neq 1, fvnctionem qvae cviqve nvmero reali xx potentiam axa^x attribvit stricte crescentem esse si a>1a > 1, et stricte decrescentem si 0<a<10 < a < 1; et, per theorema de copia imaginvm fvnctionis exponentialis, ibi ope completvdinis nvmerorvm realivm et inaeqvalitatis Bernovllianae demonstratvm, eam praeterea qvemqve valorem positivvm exacte semel assvmere, ideoqve biiectionem esse ex R\mathbb{R} in intervallvm (0,∞)\ioo{0}{\infty}. Per theorema de inversa fvnctionis stricte monotonae, in lectione de operationibvs fvnctionvm demonstratvm, haec fvnctio inversam habet. Qvod si ita est, pro qvoqve nvmero positivo xx exstat vnicvs exponens yy talis vt ay=xa^y = x, et definitio infra posita sensvm habet.

Sint a>0a > 0, cvm a≠1a \neq 1, et x>0x > 0. Logarithmvs basis aa nvmeri xx, qvi scribitvr log⁡ax\log_a x, est vnicvs nvmervs realis yy talis vt ay=xa^y = x. Fvnctio log⁡a\log_a, in (0,∞)\ioo{0}{\infty} definita et valores reales habens, est inversa exponentialis basis aa.
log⁡ax=y↔ay=x\log_a x = y \leftrightarrow a^y = x

Logarithmvs igitvr qvaestioni de exponente respondet: log⁡ax\log_a x est nvmervs ad qvem aa elevandvs est vt xx obtineatvr. Ita, cvm sit 23=82^3 = 8, 3−2=193^{-2} = \frac{1}{9}, 103=100010^3 = 1000 et, secvndvm lectionem de potentiis exponentis rationalis et radicalibvs, 412=4=24^{\frac{1}{2}} = \sqrt{4} = 2, habetvr:

log⁡28=3,log⁡319=−2,log⁡101000=3,log⁡42=12\log_2 8 = 3, \qquad \log_3 \frac{1}{9} = -2, \qquad \log_{10} 1000 = 3, \qquad \log_4 2 = \frac{1}{2}

Cvm log⁡a\log_a et exponentialis basis aa inter se inversae sint, vtraqve dissolvit qvod altera facit: prima identitas valet pro omni xx positivo, secvnda pro omni yy reali.

alog⁡ax=x,log⁡a(ay)=ya^{\log_a x} = x, \qquad \log_a\left(a^y\right) = y

Praesertim ex a0=1a^0 = 1 et a1=aa^1 = a colligvntvr log⁡a1=0\log_a 1 = 0 et log⁡aa=1\log_a a = 1 pro omni basi. Neqve nihilvm neqve nvmeri negativi logarithmvm habent, qvia omnis potentia basis positivae positiva est: nvllvs exstat yy talis vt 2y=−42^y = -4. Dvae bases nomen proprivm habent. Basis ee, nvmervs in lectione de nvmero e et exponentiali natvrali investigatvs, dat logarithmvm natvralem, qvi scribitvr ln⁡x\ln x; basis 1010 dat logarithmvm decimalem, qvi scribitvr log⁡10x\log_{10} x. In mvltis libris et calcvlatricibvs scribitvr log⁡x\log x sine basi, modo pro natvrali, modo pro decimali; in hoc cvrsv basis semper scribitvr.

ln⁡x=y↔ey=x,log⁡10x=y↔10y=x\ln x = y \leftrightarrow e^y = x, \qquad \log_{10} x = y \leftrightarrow 10^y = x

Logarithmi a Ioanne Nepero anno 1614 editi svnt, et Henricvs Briggs pavcis post annis primas tabvlas logarithmorvm decimalivm compvtavit. Plvs qvam tria saecvla ad calcvlos astronomiae, navigationis et artis machinalis contrahendos inserviervnt, qvia, vt infra demonstratvr, prodvcta in svmmas convertvnt. Delineatio logarithmi ex delineatione exponentialis obtinetvr per theorema de delineatione fvnctionis inversae, ex eadem lectione de fvnctione inversa: delineatio fvnctionis biiectivae et delineatio inversae eivs symmetricae svnt respectv rectae y=xy = x.

ExemplvmDelineatio logarithmi ad basim dvo

Delineationem fvnctionis log⁡2\log_2 ex delineatione exponentialis 2x2^x constrvimvs, eam respectv rectae y=xy = x reflectentes.

Demonstratio

  1. 2−1=12,20=1,21=2,22=4\resaltar{2^{-1}} = \frac{1}{2}, \quad \resaltar{2^0} = 1, \quad \resaltar{2^1} = 2, \quad \resaltar{2^2} = \resaltar{4}

    Aliqvot pvncta delineationis fvnctionis y=2xy = 2^x compvtamvs: pro xx aeqvali −1-1, 00, 11 et 22, potentia valet 12\frac{1}{2}, 11, 22 et 44.

  2. log⁡212=−1,log⁡21=0,log⁡22=1,log⁡24=2\dato{p}{\resaltar{\log_2 \frac{1}{2}} = -1, \quad \log_2 1 = 0, \quad \log_2 2 = 1, \quad \resaltar{\log_2 4} = 2}

    Ex definitione, qvaeqve aeqvalitas 2y=x2^y = x dicit esse log⁡2x=y\log_2 x = y: pvncta delineationis logarithmi obtinentvr coordinatis permvtatis.

  3. y=log⁡2x↔x=2y\resaltar{y = \log_2 x} \leftrightarrow \resaltar{x = 2^y}

    Idem de omnibvs pvnctis valet: pvnctvm (x,y)(x, y) in delineatione fvnctionis log⁡2\log_2 iacet si et solvm si (y,x)(y, x) in delineatione fvnctionis 2x2^x iacet. Ambae delineationes symmetricae svnt respectv rectae y=xy = x, et delineatio fvnctionis log⁡2\log_2 est reflexio delineationis exponentialis.

  4. x∈(0,∞)→log⁡2x∈Rx \in \resaltar{\ioo{0}{\infty}} \rightarrow \log_2 x \in \resaltar{\mathbb{R}}

    Reflexione dominivm et copia imaginvm permvtantvr: logarithmvs solvm pro nvmeris positivis definitvs est et omnes valores reales assvmit. Axis xx, ad qvem delineatio fvnctionis 2x2^x sinistrorsvm accedit, fit axis yy, ad qvem delineatio fvnctionis log⁡2\log_2 prope 00 accedit.