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
= 916 = 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. x3
 0.001 = 0 n‡j, x2
-Gi gvb- [35Zg wewmGm]
K) 100 L) 10 M) 10
1
N) 100
1
DËi : K
mgvavb:
x3
 0.001 = 0 ev, x3
= 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 (85) [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)
= 21 (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|