সূচক ও লগারিদম

BDwbU cuvP c„ôv 79

m~PK I jMvwi`g

Indice and Logarithm

f~wgKv

Introduction

m~PK I jMvwi`‡gi mvnv‡h¨ MvwYwZK wnmve I mgm¨v mgvavb A‡bK †ÿ‡Î mnRZi nq| m~P‡Ki mvnv‡h¨ A‡bK eo msL¨v ev †QvU msL¨v‡K ïaygvÎ GKwU cÖZx‡Ki mvnv‡h¨ cÖKvk Kiv hvq| †hgb: 100000 †K †jLv hvq10<sup>5</sup>| GKB fv‡e jMvwi`‡giI cÖ‡qvM i‡q‡Q| †hgb: RbmsL¨v e„w×, Pµe„w× gybvdvi wnmve BZ¨vw` †ÿ‡Î m~PK Ges jMvwi`g e¨envi Kiv nq| Rb †bwcqvi (John Napier 1550-1617) me© cÖ_g jMvwi`g Avwe®‹vi K‡ib| Rb †bwcqvi (Ag~j`) f~wgi Dci wfwË K‡i jMvwi`g Avwe®‹vi K‡ib Ges GwU‡K e wfwËK jMvwi`g ev ¯^vfvweK jMvwi`g ejv nq| cieZ©x‡Z Rb

†bwcqv‡ii mgmvgwqK MwYZwe` †nbwi weªMm (Henry Briggs 1561-1630) g~j` 10 wfwË jMvwi`g Avwe®‹vi I e¨envi K‡ib hv mvaviY MYbvq e¨envi Kiv nq|

BDwbU mgvwßi mgq BDwbU mgvwßi m‡e©v”P mgq 2 w`b

g~L¨ kã

g~j`, Ag~j`, m~PK, g~j, eM©g~j, n Zg g~j, msL¨vi ˆeÁvwbK ev Av`k©i~c, c~Y©K, AskK, jMvwi`g, ¯^vevweK ev cÖvK…wZK jMvwi`g, mvaviY jMvwi`g BZ¨vw`|

G BDwb‡Ui cvVmg~n cvV 5.1: m~PK cvV 5.2: jMvwi`g

5

m~PK

Indices D‡Ïk¨

G cvV †k‡l Avcwb-

 g~j` m~PK Kx Zv e¨vL¨v Ki‡Z cvi‡eb;

 abvZ¥K I FYvZ¥K c~Y©-mvswL¨K m~PK e¨vL¨v I cÖ‡qvM Ki‡Z cvi‡eb;

 m~P‡Ki m~Îvewj eY©bv I Zv cÖ‡qvM K‡i mgm¨v mgvavb Ki‡Z cvi‡eb;

 n Zg g~j‡K m~PK AvKv‡i cÖKvk Ki‡Z cvi‡eb|

m~PK

Indices

m~PK A_© kw³ ev Power| †Kvb msL¨v‡K H msL¨v Øviv GKvwaKevi ¸Y Ki‡j m~PK bv‡gi GKwU ms‡KZ Øviv G‡K cÖKvk Kiv nq| hw` x †h‡Kv‡bv ev¯Íe msL¨v Ges n ¯^vfvweK msL¨v nq Z‡e xGi µwgK ¸Y, A_©vr

x x

x x

x   ... (ⁿmsL¨K evi ^x) †K xⁿ AvKv‡i †jLv nq| GLv‡b, ⁿ n‡jv m~PK ev NvZ Ges x †K wfwË ejv nq|

m~PK ïay abvZ¥K c~Y©msL¨vB bq, FYvZ¥K c~Y©msL¨v ev abvZ¥K fMœvsk ev FYvZ¥K fMœvskI n‡Z cv‡i|

A_©vr wfwË Ix∈ 𝐑 ev¯Íe msL¨vi †mU) Ges m~PK nQ (g~j` msL¨vi †mU) Gi Rb¨ msÁvwqZ| Z‡e we‡kl †ÿ‡Î N

n (¯^vfvweK msL¨vi †mU) aiv nq|

m~P‡Ki ag©vewj (Rules for Indices) g‡b Kiæb, xR Ges m,nN

1. hw` mn nq, Z‡e x^mxⁿn‡e|

wecixZµ‡g x^mxⁿ n‡j, mn n‡e|

2. Avevi, hw` xy nq, Z‡e x^m  yⁿ n‡e|

wecixZµ‡g x0,y0n‡j, hw` x^m yⁿ nq, Zvn‡jx y n‡e|

m~P‡Ki m~Îvewj (Formulae for Indices) g‡b Kiæb, xR Ges ^m,nN

1. x^mxⁿ x^mn

2. _n ^{m n}

m

x x

x  _ hw`, mn 3. _{n m n}

m

x x x

 1

hw`, mn 4. ^m_n 1

x

x hw`, mn 5.

 

^{xm n xmn 6. (x.y)mxmym}

7. m

m m

y x y x 



 



 8. ^m

m x

x

 

1 , †hLv‡b, x0

9. _m x ^m x

 

1 , †hLv‡b, x0 10. x⁰ 1

cvV-5.1

BDwbU cuvP c„ôv 81

n Zg g~j jÿ¨ Kiæb,

2 2 1 2 1 2 1

3 3

3 













Avevi, 3 3 3 3 ² 3

2 1 2 1 2 1 2 1 2 1







 ^{ } 3 3

2 2 1

 



 





2 1

3 Gi eM© (wØZxq NvZ) 3 Ges 3Gi eM©g~j (wØZxq g~j) ²

1

3

2 1

3 †K eM©g~‡ji wPý Gi gva¨‡g 3 AvKv‡i †jLv nq|

Avevi, jÿ¨ Kiæb-

3 3 1 3 1 3 1 3 1

3 3 3

3 













Avevi, 3 3 3 3 3 ³ 3

3 1 3 1 3 1 3 1 3 1 3 1 3 1









 ^{   3 3}

3 3 1

 













3 1

3 Gi Nb (Z…Zxq NvZ)3 Ges 3Gi Nbg~j (Z…Zxq g~j) ³

1

3

3 1

3 †K Nbg~‡ji wPý ³ Gi gva¨‡g ³ 3 AvKv‡i †jLv nq|

n Zg g~‡ji †ÿ‡Î

n n

n

n x x x

x

1 1

1 1

...





 [ⁿ msL¨K xⁿ

1

Gi µwgK ¸Y]

n

xn











1

Avevi, xⁿ xⁿ xⁿ xⁿ

1 1

1 1

...







n n

x

  1

[m~P‡K nmsL¨K

n

1 Gi †hvM] x x x

n n 





 ¹

xn 1

Gi n Zg NvZ x Ges x Gi n Zg g~j xⁿ

1

 A_©vr xⁿ

1

Gi nZg NvZ x x

n n 











1

Ges xGi nZg g~j

 

x ⁿ¹ xⁿ¹ ⁿ x xGi n Zg g~j †K ⁿ xAvKv‡i †jLv nq|(⁷

3)⁵× (⁷

3)⁻⁵

 D`vniY 1: gvb wbY©q Kiæb: (K) ₉

7

5

5 (L) (⁷

3)⁵× (⁷

3)⁻⁵ mgvavb: (K)

25 1 5 5 1 5 5

5

2 2 9 7 9

7  ^  ^   (L) (⁷

3)⁵× (⁷

3)⁻⁵= (⁷

3)⁵⁻⁵= (⁷

3)⁰ = 1

D`vniY 2:mij Kiæb: (K) 7

7 . 7 ³

3 2

, (L)

2 2

2 . 4 2

2 1 4









 n

n n

mgvavb: (K) †`Iqv Av‡Q, 7

7 . 7 ³

3 2

2 1 3 1 3 2

7 7 .

7

2 1 3 1 3 2

7 ^{ }

 7^23121





2 1 1

7^



2 1

7  7

(L) †`Iqv Av‡Q,

2 2

2 . 4 2

2 1 4









 n

n n

2 2 . 2

2 . 2 . 4 2 . 2

2 4



 ⁿ _n  ⁿ

2 2 . 2

2 . 8 2 . 16

2 n

n n



2 . 2

) 1 2 ( 2 . 8

n

n 

 2

8 4

D`vniY 3: mij Kiæb:

 

x x x

x

8 2 16

4 9

1 1

2











mgvavb: †`Iqv Av‡Q,

 

x x x

x

8 2 16

4 9

1 1

2











 

^x

 

^x

 

^x

x 3 1 1

4

2 2 2

2 2 2

2 3





 

 

x x x

x 3 1 4 4

4 2

2 2 2

2 3





 _  _{ 4} ²₄ ⁴_{4 1} 2 2

2 3



 

 _x  ^x_x

2 4

4 4 4 1

3 2

2 2 2 2

x

x x

 

   2 (2 2) 2 3

4 4

4 2



 _x  ^x

14 9 ) 2 16 (

3² 

 

D`vniY 4: mij Kiæb: _{m n m}

n n m n m m

15 10 6

6 5 3 2

3 1

1 3 2

3























mgvavb: †`Iqv Av‡Q, _{m n m}

n n m n m m

15 10

6

6 5 3

2

3 1

1 3 2

3























m n

m

n n

m n m m

) 3 5 ( ) 2 5 ( ) 3 2 (

) 3 2 ( 5 3

2

3 1

1 3

2 3















 ^ _ ^{ } _^{ }

3 2 3 1 1

1 1 3 3

2 3 5 3 2

2 3 5 2 5 3

m m n m n n n

m m n n m m

     

   

   

     

m n n m m m n n m n

m n

m              

2 ^{3 1 1 3} 3^{2 1 1} 5 ^{3 3}

1 5 3 2⁰ ⁰ ⁰ 



BDwbU cuvP c„ôv 83

D`vniY 5: hw` ^{yy y }

 

^{y y y}nq Zvn‡j cÖgvY Ki‡Z n‡e †h,

4

 9 y

mgvavb: †`Iqv Av‡Q, ^{yy y }

 

^{y y y⇒yy y} _^y









 ^



2 1 1

2 1 1

⇒

y

y y

y 















2 1 2

2 1 2

⇒y^y y^2y

3

2 3

 ⇒y y

2

2 3

3



⇒ y y

y y 1

2 3

2 1

3





 , [Dfq c‡ÿ

y

1 Øviv ¸Y K‡i]

⇒ 2

3

1 2 3

 

y ⇒

2

2 3

2 3





y ⇒

2

2 3

1



y ⇒

2 2 2 1

2 3



 















y [Dfq c‡ÿ eM© K‡i]

4

9

y (cÖgvwYZ)|

D`vniY 6: †`Lvb †h, . . 1

1 1

1

 



 



 



 



 



 



 ^ca

a bc c c ab b b a

x x x

x x

x

mgvavb: L.H.S =

ca a bc c c ab b b a

x x x

x x

x

1 1

1

.

. 

 



 



 



 



 





ca a ca

c

bc c bc b

ab b ab

a

x x x x x

x . .



c a

b c

a b

x x x x x x

1 1

1 1

1 1

.

 .

c b a

c b a

x x x

x x x

1 1 1

1 1 1

. .

.

 . 1= ^R.H.S

weKí c×wZ: L.H.S =

ca a bc c c ab b b a

x x x

x x

x

1 1

1

.

. 

 



 



 



 



 





     

^{xab ab1.xbc bc1.xca ca1}



ca a c bc

c b ab

b a

x x x







 . .

ca a c bc

c b ab

b a

x

 

 





     

abc a c b c b a b a c

x













abc ab bc ac ab bc ac

x













0 1

x = R.H.S

D`vniY 7: hw` pqr1 nq Zvn‡j †`Lvb †h, 1

1 1 1

1 1

1

1 1

1 



 



 



 p q^ q r^ r p^

mgvavb: †`Iqv Av‡Q, pqr1pqr^1

L.H.S = _{1 1 1}

1 1 1

1 1

1





   



 



 p q q r r p

1

1 1

1 1

1 1

1



   



 



 p q q pq r p [r^1pq]

r p pq q

p q 1

1 1 1

1 1 1

1





 

 







p rp pq p

q q

pq

q 1

1 1

1 1

1



 



 



 

1 1

1

1   



 



 

rp p

p pq

q pq

q q

q pqr pq

pq pq

q pq

q q



 



 



 

1 1

1 [Z…Zxq As‡ki je I ni‡K 𝑝 Øviv ¸Y K‡i]

q pq

pq pq

q pq

q q



 



 



 

1 1

1

1 [†h‡nZz, pqr1]

= 1

1 1







  pq q

pq

q = R.H.S

1 1 1 1

1 1

1

1 1

1 



 



 



  _{  }

p r r

q q

p (cÖgvwYZ)

D`vniY 8: mgvavb Kiæb: 4^1x 4^1x 10

mgvavb: †`Iqv Av‡Q, 4^1x 4^1x 10

⇒ 4¹.4^x4¹.4^x 10

⇒ 10

4 . 1 4 4 .

4¹  ¹ 

x x

⇒ 1 10

4

4  

y y [g‡b Kiæb, 4^x y]

⇒ 4y² 410y [Dfq c‡ÿy Øviv ¸Y K‡i]

⇒ 4y² 10y40

⇒ 4y² 8y2y40

⇒ 4y(y2)2(y2)0

⇒ (y2)(4y2)0

y2 A_ev,

2

1 y GLb,y2 n‡j, 4^x 2

⇒ 2^2x 2

⇒ 2x1

2

1

x wb‡Y©q mgvavb,

2

1

 x

Avevi

2

1

y n‡j,

2 4^x 1

⇒ 2^2x 2^1,

⇒ 2x1

2

1



x

mvims‡ÿc:

 hw` x †h‡Kv‡bv ev¯Íe msL¨v Ges ⁿ ¯^vfvweK msL¨v nq Z‡e ^x Gi µwgK ¸Y, A_©vr ^x^x^x...^x (ⁿ msL¨K evi ^x) †K ^xn AvKv‡i †jLv nq| GLv‡b, n n‡jv m~PK ev NvZ Ges ^x †K wfwË ejv nq|

BDwbU cuvP c„ôv 85

jMvwi`g

Logarithm D‡Ïk¨

G cvV †k‡l Avcwb-

 jMvwi`g Kx Zv e¨vL¨v Ki‡Z cvi‡eb;

 †Kv‡bv m~PKxq dvskb‡K jMvwi`‡gi mvnv‡h¨ cÖKvk Ki‡Z cvi‡eb;

 jMvwi`‡gi m~‡Îi mvnv‡h¨ wewfbœ mgm¨v mgvavb Ki‡Z cvi‡eb|

jMvwi`g

Logarithm

Logos Ges Arithmas bvgK `ywU wMÖK kã n‡Z Logarithm kãwUi DrcwË n‡q‡Q| Logos A_© Av‡jvPbv Ges

Arithmas A_© msL¨v | A_©vr msL¨v wb‡q Av‡jvPbv| jM Gi gva¨‡g a^x NGi Avi GKwU Abyiƒc cÖKv‡ki gva¨g n‡jv Log_aN x| f~wg ev base a-Gi Dci m~PK DwbœZ n‡j djvdj n‡e N| myZivs, †Kvb msL¨vi GKwU cÖ`Ë wfwËhy³ jMvwi`g n‡”Q H m~PK ev NvZ ev cÖ`Ë wfwËi Dci DwbœZ Ki‡j msL¨vwU cvIqv hv‡e| †hgb: hw`a^x b

nq, †hLv‡b a0Ges a1Z‡e x†K ejv nq b Gi 𝑎 wfwËK jMvwi`g, A_©vrxlog_ab|

wecixZµ‡g, hw`xlog_ab n‡j a^x b n‡e| G‡ÿ‡Î bmsL¨vwU‡K 𝑎 Gi mv‡c‡ÿxGi cÖwZjM (Anilogarithm)

e‡j Ges †jLvi c×wZ banti log_a x_.

jMvwi`‡gi m~Îvejx (Formulas of Logarithm)

m~Î 1: log_aa1 Ges log_a10 m~Î 2: log_a(MN)log_aMlog_a N

m~Î 3: M N

N M

a a

a log log

log  



 



 m~Î 4: log_a(M)^N Nlog_aM

m~Î 5: log_aM log_bMlog_ab

m~Î 6:

M a

M

a log

log  1 or log_aMlog_Ma1

m~Î 7: log_ablog_xblog_a x

m~Î 8:

a b b

x x

a log

log log

m~Î 9: x^logay y^logax, †hLv‡b, x,y0

D`vniY 𝟏: gvb wbY©q Kiæb: (K) log₁₀100 (L) log 81

3

mgvavb: (K) log₁₀100log₁₀(10)³3log₁₀103.13 [log_aa1]

(L) ^{log 81 log 3 log}

   

^{3 log}

^{ }

^{3 8log} 3

^{ }

^{3 8} 8

3 2 4

3 4

3

3      [log_aa1]

jMvwi`‡gi e¨envi (Use of Logarithms): ‰`bw›`b Rxe‡b jMvwi`‡gi e¨envwiK Z_v e¨emvwqK cÖ‡qvM i‡q‡Q|

Avi GRb¨ we‡kl ai‡bi MYbv Kv‡h© jMvwi`‡gi cÖ‡qvRb Abyf~Z n‡q _v‡K| wb‡¤œ jMvwi`‡gi e¨envwiK cÖ‡qvM D‡jøL Kiv n‡jv:

1. m~PK ev NvZ wewkó gv‡bi MYbv jMvwi`‡gi mvnv‡h¨ `ªæZ I mn‡R Kiv hvq|

2. Pµe„w× my` msµvšÍ mgm¨v mgvav‡b jMvwi`g e¨eüZ n‡q _v‡K|

cvV-5.2

3. ARvbv my‡`i nvi Ges mgqKvj wbY©‡qi Rb¨ jMvwi`g e¨envi Kiv nq|

4. µgn«vmgvb †Ri c×wZ‡Z AeP‡qi cwigvY, m¤úwËi RxebKvj, AeP‡qi nvi cÖf…wZ wbY©‡q jMvwi`g Gi cÖ‡qvM i‡q‡Q|

5. exgv wKw¯Í wba©vi‡YI jMvwi`‡gi e¨envi jÿ¨ Kiv hvq|

6. AwZ eo Ges RwUj msL¨vi ¸Y, fvM cÖf„wZ MYbvi †ÿ‡Î jMvwi`g Gi e¨envi KvR‡K mnR K‡i †`q|

7. GKvwaK dvsk‡bi ¸Ydj wbY©q Ges AbycvZ m¤^wjZ dvsk‡bi wewfbœ MYbv Kvh© mn‡R Kivi Rb¨ jMvwi`g Gi e¨envi Kiv nq|

8. RbmsL¨v e„w×i nvi wbY©q, cÖe„w×i nvi wbY©q QvovI A_©bxwZi wewfbœ MvwYwZK Kvh©µ‡g jMvwi`g Gi e¨envi jÿ¨bxq|

jMvwi`‡gi cÖKvi‡f` (Types of Logarithm)

jMvwi`g `yB cÖKvi| h_v: 1. ¯^vfvweK ev cÖvK…wZK jMvwi`g 2.mvaviY jMvwi`g

1. ¯^vfvweK ev cÖvK…wZK jMvwi`g (Naal/ Napierian/ ‘e’ base Logarithm): MwYZwe` Rb †bwcqvi cÖ_g e wfwËK jMvwi`g Avwe®‹vi K‡ib Zv cÖvK…wZK jMvwi`g| cÖvK…wZK jMvwi`g Gi wfwË n‡jv e| ZvB GwU‡K e wfwËK jMvwi`g ev bvg Abymv‡i †bwcqvi jMvwi`g ejv nq| e Gi gvb n‡jv: ... 2.71828

! 3 1

! 2 1

! 1

11   

e | G jMvwi`g‡K

eN

log A_ev lnN(LynnN co‡Z nq) cÖZx‡Ki mvnv‡h¨ cÖKvk Kiv nq|

2. mvaviY jMvwi`g (Common Logarithm/ Briggsian/10 base Logarithm)

10wfwËK jMvwi`g‡K mvaviY ev weªMmxq jMvwi`g e‡j| 10 wfwËK jMvwi`g †K ïay log<sub>10</sub>N Øviv wb‡`©k Kiv hvq|

A_©vr, mvaviY MYbvi Rb¨ Rb¨ †bwcqv‡ii mgmvgwqK MwYZwe` †nbwi weªMm (Henry Briggs 1561-1630) G jMvwi`g e¨envi K‡ib|mvaviY jMvwi`‡gi †ÿ‡Î wfwË †jLv bv _vK‡jI 10 †K wfwË aiv nq|

D`vniY 𝟐: jMvwi`‡gi gvb wbY©q Kiæb:

(K) log 64Gi wfwËK 2 2 (L) log ¹

324Gi wfwËK 3 2

mgvavb: (K) log 64 Gi wfwËK 2 2 𝑙𝑜𝑔_2√264 = 𝑙𝑜𝑔_2√22⁶

= 𝑙𝑜𝑔_2√2(2 × √2)⁴ = 4𝑙𝑜𝑔_2√22 × √2 = 4 × 1

(L) log ¹

324Gi wfwËK 3 2

 

⁴

4 2

1 1 1

log log log

324 3 2 3 2

 

 

 

⁴

2

3 3 2

log  ^



2 3 log 4 _{3 2}





4 1 4 



 = 4

BDwbU cuvP c„ôv 87

vniY 𝟑: hw`

3 31

log 27 x nq, Z‡exGi gvb wbY©q Kiæb|

mgvavb: †`Iqv Av‡Q,

3 10 3 31 log 27 x 

myZivs,

 

^{27 103}

 

^{33 21 103}









 x

125 3

3 ^{3 5}

10 2 3





 ^

wb‡Y©q x125

D`vniY 𝟒: cÖgvY Kiæb, log 2 2log 5 log 3 2log 7 147

log₁₀ 50  ₁₀  ₁₀  ₁₀  ₁₀

mgvavb: evgcÿ =

147

log₁₀ 50 = ₂

2 10

10 3 7

5 log 2 7 7 3

2 5 log 5



 









=

 

10



²



2

10 2 5 log 3 7

log   



10 10 ²



2 10

102 log 5 log 3 log 7

log   



7 log 2 3 log 5 log 2 2

log₁₀  ₁₀  ₁₀  ₁₀



= Wvbcÿ (cÖgvwYZ) D`vniY 𝟓: cÖgvY Kiæb †h, x^logay  y^logax

mgvavb: g‡b Kiæb, log_a y pGes log_a xq

myZivs, a^p yGes a^q x

 

^{ap q  yq}

 , [Dfq c‡ÿ q NvZ wb‡q]

ev,a^pq y^q......(i)

Avevi,

 

^{aq p xp [Dfq c‡ÿ p} NvZ wb‡q]

ev, a^pq x^p.....(ii)

( ) I (ii) bs †_‡K cvB, x^p  y^q

x

y _a

a y

x^log  ^log

 (cÖgvwYZ)

D`vniY 𝟔: cÖgvY Kiæb, 6 1 log 3 log 5

log₅ ²  ₃ ⁴ ₁₀ 

mgvavb: L.H.S.log₅5²log₃3⁴log₁₀1 0

3 log 4 5 log

2 ₅  ₃ 



1 4 1 2  



2 4 6 R.H.S.

   

log₅5²log₃3⁴log₁₀16(cÖgvwYZ)

D`vniY 7: cÖgvY Kiæb †h, 3log2 10

log 16 log

8 log

4 9

3 



mgvavb:

10 log 16 log

8 . log

. .

4 9

3

  S H L

4 log

10 log 9 log

16 log

3 log

8 log

e e e

e e e





10 log

4 log 16 log

9 log 3 log

8 log

e e e

e e

e  



10 log

2 log 2 log

3 log 3 log 2

2

log ²

4 2 3

e e e

e e

e  



10 log

2 log 2 2 log 4

3 log 2 3 log

2 log 3

e e e

e e

e  



1 2 log 2 2 log 4

3 log 2 3 log

2 log

3 _e

e e e

e  



. . . 2 log

3 _e RH S



2 log 10 3 log 16 log

8 log

4 9

3 

  (cÖgvwYZ)

mvims‡ÿc:

 †Kvb msL¨vi GKwU cÖ`Ë wfwËhy³ jMvwi`g n‡”Q H m~PK ev NvZ ev cÖ`Ë wfwËi Dci DwbœZ Ki‡j msL¨vwU cvIqv hv‡e| †hgb: hw`a^x b nq, †hLv‡b a0Ges a1Z‡e x†K ejv nq b Gi 𝑎 wfwËK jMvwi`g, A_©vrxlog_ab|

 m~Î 1: log_a(MN)log_aMlog_a N

 m~Î 2: M N

N M

a a

a log log

log  



 





 m~Î 3: log_a(M)^N Nlog_aM