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Sucok o logaridom [www.onlinebcs.com]
1.
Bb‡mckb wU‡gi D‡`¨v‡M
AbjvBb wewmGm wcÖwjwgbvwi †Kvm© hviv MwY‡Z `ye©j Zviv wb‡Pi 15wU UwcK fvjfv‡e Ki‡jB wcÖwjwgbvwi cixÿvi 15wU As‡Ki g‡a¨ Kgc‡ÿ 10wU AsK cvi‡eb| ev¯Íe msL¨v j.mv.¸. I M.mv.¸ kZKiv mij I †hŠwMK gybvdv AbycvZ-mgvbycvZ jvf-ÿwZ exRMvwYwZK m~Îvejx eûc`x Dccv`¨ m~PK I jMvwi`g mgvšÍi I ¸‡YvËi aviv ‡KvY, wÎfzR PZzf©yR ‡mU web¨vm-mgv‡ek m¤¢ve¨Zv cÖ‡Z¨KwU Uwc‡Ki m‡e©v”P 10wU K‡i wbqg A_©vr 10 15 = 150 wbqg wk‡L †dj‡jB 10 b¤^i wbwðZ| Avgiv GB 150wU wbqg Avcbv‡`i wkwL‡q w`e Bb-kv-Avjøvn| AvR‡Ki Av‡jvPbvi UwcK t m~PK bZzb wm‡jev‡mi Av‡jv‡K wewmGm wcÖwjwgbvwi cixÿvi cÖkœ we‡kølY UwcK 35Zg 36Zg 37Zg 38Zg 40Zg m~PK I jMvwi`g 1 1 1 1 2 1 1 1 1 cÖ‡qvRbxq m~Îvejx a. am . an = am+n (¸‡Yi mgq cvIqvi †hvM nq) b. am an = an am =am-n (fv‡Mi mgq cvIqvi we‡qvM nq) GLv‡b, m hw` n Gi †P‡q eo (m ˃ n ) nq Zvn‡j Dc‡ii m~ÎwU cÖ‡hvR¨ nq| hw` m ‡QvU nq n Gi †P‡q (m < n) Zvn‡j wb‡Pi m~ÎwU cÖ‡hvR¨ n‡e- c. a0 =1 [‡hLv‡b a 0 A_©vr a Gi gvb Aek¨B 0 Gi Kg ev ‡ewk n‡e]
2.
Bb‡mckb wU‡gi D‡`¨v‡M
AbjvBb wewmGm wcÖwjwgbvwi †Kvm© ‡h wZbwU m~Î †kLv n‡jv, †mB wZbwU m~Î †_‡K weMZ eQ‡i Avmv A¼ †kLv hvK---- 1. am . an = am+n KLb n‡e? (14Zg wewmGm) K. m abvZ¥K n‡j L. n abvZ¥K n‡j M. m I n DfqB abvZ¥K N. m abvZ¥K I n FYvZ¥K DËi : M mgvavb t [GUv c„_g c„ôvq DwjøwLZ m~Îmg~‡ni cÖ_g m~ÎwU (a)| GB AsKwU mgvav‡bi Rb¨ Avcbv‡K wP‡ýi †hvM-we‡qv‡Mi mvaviY wbqgm~n Rvb‡Z n‡e| †hgb : + + = +, + - = -, - - = + cÖf…wZ| Gevi cÖkœwU jÿ¨ Kiv hvK, - m+(-n) = -m-n, m+(-n) = m-n . Zvn‡j †`Lv hv‡”Q ïay m I n abvZ¥K n‡jB WvbcÿwU cvIqv hvq| 2. a0 = KZ? (31 Zg wewmGm, gvbwmK `ÿZv) a. 1 b. 0 c. a d. 2 Ans : a mgvavb: [ûeû c~‡e©i c„ôvq DwjøwLZ 3 bs m~ÎwU (c) GwU] †h †Kvb msL¨vi Dci cvIqvi 0 n‡j Zvi gvb 1 nq| Z‡e msL¨vwU Aek¨B 0 Gi †P‡q eo ev †QvU n‡Z n‡e| †hgb: 1 ba ab 0 , A_©vr Ggb nIqv hv‡e bv| 3. a5 a a-6 = KZ? a.0 b. 1 c. 3 d. 2 Ans : b mgvavb : [cÖ_g c„ôvq DwjøwLZ cÖ_g I wØZxq m~‡Îi (a & c) gva¨‡g Ki‡eb] cÖ_‡g cvIqv‡ii KvR Kiv hvK| ¸‡Yi mgq cvIqvi †hvM nq| Gevi cÖ_g c„ôvi 3 bs m~ÎwU cÖ‡qvM Ki‡Z n‡e| 02 bs As‡K Gi we¯ÍvwiZ e¨vL¨v cv‡eb|
3.
Bb‡mckb wU‡gi D‡`¨v‡M
AbjvBb wewmGm wcÖwjwgbvwi †Kvm© d. a = a 2 1 (iæU D‡V †M‡j cvIqviiƒ‡c 2 1 nq| Gevi wb‡Pi m~ÎUv †`Lyb) 4. √ √ Gi gvb KZ? [26Zg wewmGm] K. 20 L. 60 M. 225 15 DËi : M mgvavb : d bs Gi NUbvwU jÿ¨ Kiæb| (√ √ ) = e. q a = q 1 a (iæU D‡V †M‡j cvIqviiƒ‡c 2 1 nq| iæ‡Ui †cQ‡b q _vKvq 2 Gi RvqMvq q emj| hw` q Gi ¯’‡j Ab¨ †Kvb msL¨v _vKZ, Zvn‡j 2 Gi ¯’‡j †m msL¨vwUB emZ| Avevi √ = ) 5. 63 43 3 = KZ? [33Zg wewmGm] a.27 b.121 c.144 d.140 Ans : c mgvavb : e bs Gi NUbvwU jÿ¨ Kiæb| cvIqv‡ii Dci hw` AveviI cvIqvi _v‡K, Zvn‡j Df‡qi g‡a¨ ¸Y nq| 63 43 3 = 6 3 6 3 43 = = 32 42 = 916 = 144 f. a-n = an 1 A_©vr a-2 = 2a 1 Note : †Kvb cvIqvi gvBbvm _vK‡j Zv fMœvsk AvKv‡i wjL‡Z nq Ges gvBbvm Zz‡j je 1 Gi wb‡P cyiv msL¨vwU cvIqvi mn wjL‡Z nq) g. -p n m = p m n (fMœvs‡ki Dc‡ii cvIqvi wU gvBbvm n‡j H fMœvskwU D‡ë hvq)
4.
Bb‡mckb wU‡gi D‡`¨v‡M
AbjvBb wewmGm wcÖwjwgbvwi †Kvm© 6. (a-1 +b-1 )-1 [a o, b o] a. ba ab b. a ba c. ab ba d. b ba Ans : a mgvavb: [f Ges g bs m~Î `ywUi cÖ‡qv‡M GB AsKwU mgvavb Kie] [(a-1 +b-1 )-1 [a o, b o) < Gi A_© a I b Gi gvb 0 †_‡K eo] = 1 11 ba [ (-) cvIqvi _vK‡j fMœvsk AvKv‡i wjL‡Z nq (f bs m~‡Îi cÖ‡qvM n‡j)|] = 1 ab ba = ba ab [fMœv‡ki Dci FYvZ¥K cvIqvi _vK‡j fMœvskwU D‡ë hvq (g bs m~Îvbymv‡i FYvZ¥K cvIqvi _vKvi Kvi‡Y Zv D‡ë †Mj)] 7. y.x 1 z.y 1 x.z 1 Gi gvb KZ? K. 35 L. 1 M. -1 N. 100 DËi : L mgvavb: yx 1 . zy 1 . xz 1 . = y x 1 . z y 1 . x z 1 . = x y y z z x = z x y z x y .. = 1 =1 h. ax = ay n‡j, x = y (A_©vr `y cv‡ki wfwË wg‡j ‡M‡j `y‡Uv wfwËB Zz‡j w`‡q mvg‡b G¸‡Z nq) i. ax = bx n‡j, a = b (`y cv‡ki cvIqvi wg‡j ‡M‡j `y‡Uv cvIqviB ev` w`‡q mgvavb Ki‡Z nq) FYvZ¥K cvIqvi _vKvq Zv D‡ë †Mj|
5.
Bb‡mckb wU‡gi D‡`¨v‡M
AbjvBb wewmGm wcÖwjwgbvwi †Kvm© 8. (1000) 3 x =10 n‡j, x Gi gvb KZ? a.0 b.1 c. 3 d. 2 Ans : b mgvavb: (1000) 3 x =10 (Wv‡b 10 Av‡Q ZvB ev‡gi 1000 †KI 10 wfwË evbv‡Z n‡e) ev, (103 ) 3 x =10 ev, (10) 3 3x = 10 (GKwU wfwËi Dci `ywU cvIqvi _vK‡j `y‡UvB ¸Y Ki‡Z nq) ev, 10x = 10 ev, 10x = 101 (`y cv‡kB 10 ZvB , `ywU 10 B ev` †`qv hvq h bs m~Î Abymv‡i) x = 1 wb‡R wb‡R PP©v Kiæb| bv eyS‡j †cv‡÷ K‡g›U Kiæb| eywS‡q †`qv n‡e| 9. (x2 )3 †K x3 -Øviv ¸Y Ki‡j KZ n‡e? (Zzjv Dbœqb †ev‡W©i Kg©KZv- 97) DËi : x9 [wb‡`©kbv : cvIqv‡ii Dci cvIqvi _vK‡j `yB cvIqviB ¸Y Ki‡Z nq] 10. 1 11 32 = KZ? DËi : - 1 [Hints : f Ges g bs m~‡Îi cÖ‡qvM Kiæb] 11.hw` 3 2 64 + 2 1 625 = 3K nq, Z‡e K =? [31-Zg wewmGm] K. 2 1 9 L. 3 1 11 M. 5 2 12 N. 3 2 13 DËi : N mgvavb: 3 2 64 + 2 1 625 = 3K ev, 3 2 6 2 + 2 1 4 5 = 3K ev, 4 2 + 2 5 = 3K ev, 16 + 25 = 3K ev, 3K = 41 K = 3 2 13
6.
Bb‡mckb wU‡gi D‡`¨v‡M
AbjvBb wewmGm wcÖwjwgbvwi †Kvm© 12. 3 3 3 a Gi gvb n‡e- [33Zg wewmGm] a. 3 1 a b. 9 1 a c. 27 1 a d. 3 a Ans : a mgvavb: 3 3 3 a = 3 1 3 3 1 3 aa [e bs m~Îvbymv‡i cÖ_‡g GKUv wKDe iæU Gi Zz‡j w`‡q emv‡jI Av‡iKUv _v‡K, hv cvIqvi 3 Gi mv‡_ ¸Y nq Ges †k‡l wKDe iæU Zz‡j w`‡q msL¨v ewm‡q mgvavb Kiv n‡q‡Q|] 13. 2x + 2x + 2x + 2x Gi gvb KZ? a.2x+3 b. 2x c. 4 d. 2x+2 Ans : d mgvavb : 2x + 2x + 2x + 2x (GLv‡b 4 wU 2x Av‡Q| †hvM AvKv‡i Av‡Q, ¸‡Yi mv‡_ ¸wj‡q †dj‡eb bv| bv eyS‡j K‡g‡›U Rvbvb|) = 4.2x (4.2x ………..2x ¸Y) = 22 . 2x (`y‡Uv msL¨viB wfwË †gjv‡bvi Rb¨ 4 †K 22 wjLv n‡q‡Q) = 22+x (wfwË GKB ZvB 2 Kgb †bqv n‡q‡Q Ges ¸Y Ae¯’vq _vKvi Kvi‡Y cvIqvi ¸‡jv †hvM) = 2x+2 14.4x + 4x + 4x + 4x Gi gvb wb‡Pi †KvbwU? [33Zg wewmGm] [Help: 4. 4x ‡K fv½yb ] DËi : 22x+2 [wb‡`©kbv t 13 bs cÖ‡kœi mgvavb †`Lyb| bv cvi‡j K‡g‡›U Rvbvb|] 15. 230 + 230 + 230 + 230 = ? DËi t 232 [wb‡`©kbv t 13 bs cÖ‡kœi mgvavb †`Lyb| bv cvi‡j K‡g‡›U Rvbvb|] 16. x4 = 81 Ges x abvZ¥K n‡j x Gi gvb KZ? a. 2 b.3 c.4 d.5 Ans : b mgvavb: x4 = 81 ev, x4 = 34 ev x=3
7.
Bb‡mckb wU‡gi D‡`¨v‡M
AbjvBb wewmGm wcÖwjwgbvwi †Kvm© 17.hw` 3x+2 = 81nq, Z‡e 3x-2 = KZ? DËi t 1 [wb‡`©kbv t 16 bs cÖ‡kœi mgvavb jÿ¨ Kiæb| bv cvi‡j K‡g‡›U Rvbvb] 18. 22x+2 = 8x+3 n‡j x = KZ? DËi t -7 [wb‡`©kbv t 16 bs cÖ‡kœi mgvavb jÿ¨ Kiæb| bv cvi‡j K‡g‡›U Rvbvb] 19. hw` (25)2x+3 = 53x+6 nq, Z‡e x = KZ? [36Zg wewmGm] K. 0 L. 1 M. -1 N. 4 DËi : K mgvavb: (25)2x+3 = 53x+6 ev, (52 )2x+3 = 53x+6 ev, (5)4x+6 = 53x+6 ev, 4x+6 = 3x+6 x = 0 20. x3 0.001 = 0 n‡j, x2 -Gi gvb- [35Zg wewmGm] K) 100 L) 10 M) 10 1 N) 100 1 DËi : K mgvavb: x3 0.001 = 0 ev, x3 = 0.001 ev, 3x 1 = 1000 1 ev, x3 = 1000 ( AvovAvwo ¸Y K‡i) ev, x3 = 103 (`ycv‡k power mgvb Kivi Rb¨ ) ev, x = 10 (power wg‡j hvIqvq cvIqvi ev`) GB 10 B DËi bv, †Kbbv 10 nj x Gi gvb| wKš‘ cÖ‡kœ x2 Gi gvb †ei Ki‡Z ejv n‡q‡Q| x=10 n‡j x2 =100| 21. 2x +21-x = 3 n‡j x = KZ? [36Zg wewmGm wjwLZ:] Ges [38Zg wewmGm wcÖwj:] (K) (1,2) (L) (0,2) (M) (1,3) (N) (0,1) DËi : N Ackb †_‡K [Back Solved Method] x = 0 n‡j, 20 +21-0 = 3 1+2=3 3=3 Avevi: x = 1 n‡j, 21 +21-1 = 3 2+1=3 3=3 = Gi `ycv‡k 3 wg‡j hvIqvq (0,1) nj Kvw•ÿZ DËi| G‡ÿ‡Î me¸‡jv Ackb w`‡q †gjv‡Z nq| bv eyS‡j K‡g‡›U Rvbvb|
8.
Bb‡mckb wU‡gi D‡`¨v‡M
AbjvBb wewmGm wcÖwjwgbvwi †Kvm© 22. 2562 3x2 n‡j, x = KZ? DËi t 1 23. 5)(x a b 3x b a nq Z‡e x = KZ? [33Zg wewmGm] K. 8 L. 3 M. 5 N. 4 DËi : N mgvavb: 5)(x a b 3x b a {`ycv‡ki wfwË †gjv‡Z n‡e} ev, 5)-(x b a 3x b a {fMœvsk Dëv‡j Dc‡ii cvIqv‡ii Av‡M(-) nq|} ev, x-3 = -x+5 [wfwË wg‡j †M‡Q ZvB wfwË ev`] ev, 2x = 8 x = 4 24. If (16)2x+3 =(4) 3x+6 then x =? (&DËi t 0 | bv cvi‡j K‡g‡›U Rvbvb) 25. n 1n2n 4.5 5355 = KZ? [34 Zg wewmGm] K. 4 L. 8 M. 12 N. 16 DËi :L mgvavb: n 1n2n 54 55755 = n nn 54 57255 = n n 54 7255 = 4 32 = 8 26. 2x 1x4x 3 9.33 Gi gvb KZ ? K. 6 L. 0 M. 27 N. 6 1 DËi : K mgvavb: 2x 1x4x 3 393 . = 2x 1x24x 3 333 . = 2x 3x4x 3 33 = 2x 3x4x 33 3333 . .. = 9x3 2781x3 . )( = 9 54 = 6 27. 3.2n -4.2n-2 = KZ? K. 2n L.2n+1 M. 2n-1 N. 23 DËi : L
9.
Bb‡mckb wU‡gi D‡`¨v‡M
AbjvBb wewmGm wcÖwjwgbvwi †Kvm© mgvavb: 3.2n -4.2n-2 = 3.2n -22 . 2n-2 =3.2n -22+n-2 (2 Gi wfwË wg‡j hvIqvq Ges ¸Y Ae¯’vq _vKvq 2 Kgb wb‡q cvIqvi †hvM n‡q‡Q) =3.2n -2n =2n (3-1) (`y ivwk‡ZB 2n Av‡Q ZvB 2n Kgb) =2n .2=2n+1 28. 3mx-1 = 3amx-2 n‡j x Gi gvb KZ?[WvK Awa`߇ii wewìs Ifviwkqvi-2018] [9g-10g †kÖbx-(D”PZi MwYZ Aby: 5.3 Gi D`vniY -4 ûeû|) ] K. m 2 L. 2m M. 2 m N. 0 DËi : K mgvavb: 3mx-1 =3.amx-2 (GLv‡b †KD `ycvk †_‡K 3 ev` w`‡j fzj n‡e| KviY Wvbcv‡k 3 Gici a ¸Y Ae¯’vq Av‡Q) 3 3 1mx = amx-2 3mx-2 =amx-2 1 a 3 2mx 2mx 1 a 3 2mx 02mx a 3 a 3 mx-2 = 0 x= m 2 GKUv eB evRv‡i w`‡qB Avgv‡`i `vwqZ¡ †kl n‡q †M‡Q| GUv Avgiv g‡b Kwi bv| Avgiv g‡b Kwi, hviv eBwU msMÖn K‡i‡Q, Zv‡`i 140* Kgb cvIqv‡bvi `vwqZ¡ Avgv‡`i| †m b¤^i Kgb cvIqv‡bvi Rb¨ Avgiv me ai‡bi mv‡cvU© cÖ`vb Kie| Bb‡mckb WvB‡R÷ msMÖn K‡i Avgv‡`i MÖæ‡ci mv‡_ †kl ch©šÍ _vK‡j †mUv m¤¢e n‡e Bb-kv-Avjøvn|
10.
Bb‡mckb wU‡gi D‡`¨v‡M
AbjvBb wewmGm wcÖwjwgbvwi †Kvm© hviv MwY‡Z `ye©j Zviv wb‡Pi 15wU UwcK fvjfv‡e Ki‡jB wcÖwjwgbvwi cixÿvi 15wU As‡Ki g‡a¨ Kgc‡ÿ 10wU AsK cvi‡eb| ev¯Íe msL¨v j.mv.¸. I M.mv.¸ kZKiv mij I †hŠwMK gybvdv AbycvZ-mgvbycvZ jvf-ÿwZ exRMvwYwZK m~Îvejx eûc`x Dccv`¨ m~PK I jMvwi`g mgvšÍi I ¸‡YvËi aviv ‡KvY, wÎfzR PZzf©yR ‡mU web¨vm-mgv‡ek m¤¢ve¨Zv cÖ‡Z¨KwU Uwc‡Ki m‡e©v”P 10wU K‡i wbqg A_©vr 10 15 = 150 wbqg wk‡L †dj‡jB 10 b¤^i wbwðZ| Avgiv GB 150wU wbqg Avcbv‡`i wkwL‡q w`e Bb-kv-Avjøvn| AvR‡Ki Av‡jvPbvi UwcK t jMvwi`g bZzb wm‡jev‡mi Av‡jv‡K wewmGm wcÖwjwgbvwi cixÿvi cÖkœ we‡kølY UwcK 35Zg 36Zg 37Zg 38Zg 40Zg m~PK I jMvwi`g 1 1 1 1 2 1 1 1 1 cÖ‡qvRbxq m~Îvejx j. loga(MN) = logaM + logaN [¸Y AvKv‡i ( )_vK‡j Avjv`v K‡i †jLvi mgq Df‡qi g‡a¨ †hvM wPý (logaM + logaN) e‡m] 1. 3log2 + log5 =? a. log20 b. log40 c. log60 d. log12 Ans : b mgvavb : 3log2 + log5 = Log23 + log5 (j‡Mi wbqgvbymv‡i, log Gi mvg‡bi msL¨vwU cvIqvi n‡q hvq) = Log 8+ log 5 = Log (85) [a bs m~Îvbyymv‡i, e‡·i m~‡Îi Av‡jv‡K] = log 40 loga + logb + logc = log (abc)
11.
Bb‡mckb wU‡gi D‡`¨v‡M
AbjvBb wewmGm wcÖwjwgbvwi †Kvm© k. loga N M = loga M - loga N [fvM AvKv‡i (loga N M ) _vK‡j Avjv`v K‡i †jLvi mgq Df‡qi g‡a¨ we‡qvM wPý (loga M - loga N) e‡m] 2. loga n m = KZ? [30-Zg wewmGm] [b bs m~‡Îi cÖ‡qvM Kiæb] a. logam - logan b. logam + logan c. logam logan d. ‡KvbwUB bq Ans:a l. logaMn = n logaM (log Gi †ÿ‡Î cvIqvi mvg‡b P‡j Av‡m) 3. log39 = KZ? a. 2 b. -7 c. 5 d. 3 Ans:a mgvavb: log39 = log332 = 2 log33 (cvIqvi ïiæ‡Z P‡j hvq) = 21 (log Gi wfwËg~j Ges wfwË wg‡j ‡M‡j Zvi gvb 1 nq) = 2 4. 5log55- log525 = KZ? a. 7 b. -7 c. 5 d. 3 Ans.d mgvavb t log Gi AsK¸‡jv Kivi mgq, me mgq †Póv Ki‡eb wKfv‡e wfwËg~j I wfwˇK mgvb Kiv hvq, ‡Kbbv wfwËg~j I wfwË wg‡j †M‡j `y‡UvB ev` ‡`qv hv‡e (KviY logaa = 1), ZLb Zvi cvIqviwUB n‡e DËi| K_vwU †evSvi Rb¨ AsKwUi mgvavb Kiv hvK- 5log55 - log525 = 5log55 - = 5log55 - 2log55 [c bs m~Î †gvZv‡eK, cvIqviwU log Gi mvg‡b Avmj] = log55 (5-2) = l 3 = 3 [logaa = 1] 5. 3 Log 81 KZ? [36Zg wewmGm] a. 4 b. 27 3 c.8 d. 8 1 Ans.c mgvavb : 3 Log 81 = 3 Log 34 = √ √ = 3 Log 8 3 loga - logb = log ( )
12.
Bb‡mckb wU‡gi D‡`¨v‡M
AbjvBb wewmGm wcÖwjwgbvwi †Kvm© = 8 3 Log 3 [c bs m~Îvbymv‡i cvIqvi mvg‡b P‡j †Mj] = 1 8 = 8 6. Log66 6 = KZ? a. 2 1 b. 2 3 c. 3 2 d. 4 3 Ans : b mgvavb: GLv‡b wfwËg~j 6 Gi Dc‡i 6 Ges 6 msL¨v `ywU ¸Y Ae¯’vq Av‡Q| ZvB log66 6 = log66 2 1 6 (GLv‡b wb‡Pi wfwËg~‡ji †Kvb KvR Kiv hv‡e bv| ïay Dc‡ii Ask wb‡q KvR Ki‡Z n‡e ) = log6 2 1 1 6 (GLv‡b 6 Ges 6 `ywU wfwË wg‡j hvIqvq I ¸Y Ae¯’vq _vKvq cvIqvi †hvM ) = log6 2 12 6 = Log6 2 3 6 = 2 3 7. log264 + log28 Gi gvb KZ? a. 8 b. 7 c. 9 d. 2 Ans : c mgvavb t log264 + log28 = log226 + log223 = 6log22 + 3log22 [cvIqvi mvg‡b P‡j †Mj] = 6 + 3 = 9 d. logaa = 1 Ges loga1 = 0 [GLv‡b a ˃ 0 Ges a 1 ] 8. ‡Kvb k‡Z© loga 1 = 0 [9g -10g †kÖYxi D”PZi MwYZ- jMvwi`g Aa¨v‡qi ïiæi Av‡jvPbv] [40Zg wewmGm] K. a 0, a 1 L. a 0,a 1 M. a 0, a = 1 N. a 1, a 0 DËi : K mgvavb t loga 1 = 0 n‡e hLb a 0, a 1. A_©vr a Gi gvb 1 ev‡` 0 Gi †_‡K eo †h †Kvb msL¨v n‡Z cv‡i| e. logay = x n‡j ax = y ( wfwË cwieZ©‡bi GB m~ÎwU LyeB ¸iyZ¡c~Y© ) (‡Kvb power = †Kvb gvb ‡`qv _vK‡j log Zz‡j w`‡q H cvIqvi I gvbwU ¯’vb e`j K‡i A_©vr power Gi RvqMvi gvbwU Ges gvb Gi RvqMvq power n‡q
13.
Bb‡mckb wU‡gi D‡`¨v‡M
AbjvBb wewmGm wcÖwjwgbvwi †Kvm© hvq) A_©vr logax = b n‡j ab = x wjLv hvq| 9. logx 9 1 = -2 n‡j, x-Gi gvb KZ? a. 7 b. -7 c. 5 d. 3 Ans:d mgvavb: logx 9 1 = -2 ev, x-2 = 9 1 [cvIqvi I gvbwU ¯’vb cwieZ©b Kij e bs m~Îvbymv‡i Ges log D‡V †Mj] ev, x-2 = 2 3 1 ev, x-2 = 3-2 x = 3 10. Logx( 8 1 ) = -2 n‡j x = KZ? [38Zg wewmGmw cÖwj:] K. 2 L. 2 M. 22 N. 4 DËi : M mgvavb : Logx( 8 1 ) = -2 x-2 = 8 1 [cvIqvi I gvbwU ¯’vb cwieZ©b Kij e bs m~Îvbymv‡i Ges log D‡V †Mj] 8 1 x 1 2 x2 =8 x= 24 x = 22 11. log10 (0.001) = KZ? K. 2 L. 3 M. 2 1 N. 3 1 DËi : L mgvavb: awi, log10 (.001)= x ev, 10x = .001 [m~Îvbymv‡i cvIqvi I gvbwU ¯’vb cwieZ©b Kij e bs m~Îvbymv‡i Ges log D‡V †Mj] ev, 10x = 10-3 x = - 3 12. loga x = 1, loga y = 2 n‡j, loga z = 3 n‡j, loga z yx 23 Gi gvb KZ? [35-Zg wewmGm] K) 1 L) 2 M) 4 N) 5 DËi :M mgvavb : logay = x n‡j ax = y m~Î e¨envi K‡i mgvavb Kie|
14.
Bb‡mckb wU‡gi D‡`¨v‡M
AbjvBb wewmGm wcÖwjwgbvwi †Kvm© logax = n n‡j x = an ; logax = 1 ev, a1 =x x = a1 = a loga y = 2 ev, a2 = y y = a2 logaz = 3 ev, a3 =z z = a3 GLb, loga z yx 23 = loga 3 223 a aa ).( = logaa4 = 4logaa = 4 1 = 4 13.logx 2 3 = - 2 1 n‡j, x - Gi gvb KZ? (37-Zg wewmGm wcÖwj) (K) 9 4 (L) 4 9 (M) √ (N) √ DËi:K mgvavb: logx 2 3 = - 2 1 2 1 x = 2 3 2 1 x 1 = 2 3 (cvIqviG gvBbvm _vK‡j Zv fMœvsk AvKv‡i wjL‡Z nq) x 1 = 2 3 x = 3 2 x = 2 3 2 x = 9 4 14. √ = x n‡j x Gi gvb KZ? a. 7 b. 3 2 c. 5 d. 3 Ans:b mgvavb: √ = x ev, 40052 x (cvIqvi Ges gvb ‡K ¯’vbvšÍwiZ K‡i) ev, x 52 =16 25 (400 ‡K Ggb msL¨v w`‡q fv½‡Z n‡e †hb eM© msL¨v nq) ev, x 52 =24 52 ev, x 52 = 24 4 5 ev, x 52 = 4 52 x = 4 wb‡R wb‡R evmvq Abykxjb Kiæb| bv cvi‡j K‡g‡›U Rvbvb|
15.
Bb‡mckb wU‡gi D‡`¨v‡M
AbjvBb wewmGm wcÖwjwgbvwi †Kvm© 15.log5 3 5 = KZ? DËi t [wb‡`©kbv t b Ges c bs m~Î Abymv‡i mgvavb Kiæb| bv cvi‡j K‡g‡›U Rvbvb|] 16.log2 64 + log2 8 Gi gvb KZ ? [Z_¨-gš¿Yvj‡qi wb‡qvM-2019] DËi t 9 17. 27 1 3log KZ? DËi t -3 [Help:log 33-3 = -3] 18.log3 9 1 Gi gvb- [35Zg wewmGm] DËi t -2 [Help: log3 9 1 = 2 3 1 3log = log33-2 = -2] 19.log10x=2 n‡j x= KZ? DËi t 100 [Help: 102 = x x=100 ] 20.log10x = -2, n‡j x Gi gvb KZ? DËi t [Help:10-2 = x ev, x = 2 10 1 x= 100 1 ] 21. logx144 = 4 n‡j x = KZ? DËi t 2 3 [Help:x4 = 144 or, x4 =24 32 or,x4 = 4 32 x = 32 ] 22. √ = KZ? Help: √ = √ 24 = √ 8 )2( = 8] 23.logx324 =4 n‡j, x Gi gvb KZ? [Help: logx324 = 4 ev, x4 = 324 ev, x4 = 81 4 ev, x4 = 34 4 2 ev, x4 = 4 23 x= 23 ] 24.32 Gi 2 wfwËK jMvwi`g KZ? (13Zg wewmGm) [Help: Log232 (32 Gi 2 wfwËK jMvwi`g KZ ej‡Z †evSvq Log Gi 2 wfwËi Dci 32 emv‡j Zvi gvb KZ n‡e) ] =Log225 = 5 25.400 Gi jM 4 n‡j wfwË KZ? [Z_¨-gš¿Yvj‡qi wb‡qvM-2019] mgvavb: awi, wfwË x ; myZivs logx 400 = 4 x4 = 400 x4 = 4 52 x = 52 GB AsK KqwU fvjfv‡e ey‡S wk‡L †dj‡jB wcÖwjwgbvwi cixÿvi Rb¨ jMvwi`g wb‡q mgm¨vq co‡eb bv| Bb‡mckb WvB‡R÷ †_‡KB G¸‡jv Zz‡j †`qv n‡q‡Q| eBwU msMÖn K‡i MÖæ‡ci mv‡_B _vKzb| cÖwZwU UwcK nvZ a‡i a‡i †kLv‡bv n‡e Bbkv-Avjøvn|