SSC (Class 9-10) Math BD: নবম-দশম শ্রেণি সাধারণ গণিতঃ অনুশীলনী-৪.২ লগারিদমঃ সরল, প্রমান ও সমাধান

ssc math solutions,class 9-10 math solution bd,ssc math pdf book, download pdf ssc/nine ten,নবম-দশম শ্রেণি সাধারণ গণিতঃ অনুশীলনী-৪.২ লগারিদমঃ সরল

লগারিদমঃ সরল, প্রমান সমাধান


. মান নির্ণয় করঃ


) log381

সমাধানঃ

log381
= log334
=4. log33 [log33=1]
=4.1
=4


) log53√5

সমাধানঃ

log53√5
= log551/3
=(1/3). log55 [log55=1]
=(1/3).1
=1/3


) log42

সমাধানঃ

log42
= log4√4
= log441/2
=(1/2). log44  [log44=1]
=(1/2).1
=1/2


) log2√5400

সমাধানঃ

log2√5400
= log2√51625
= log2√52452
= log2√524(√5)4
= log2√5(2√5)4
=4. log2√52√5 [log2√52√5=1]
=4.1
=4


) log5(3√5. √5)

সমাধানঃ

log5(3√5. √5)
= log551/3.51/2
= log551/3+1/2
= log551/3+1/2
= log555/6
=(5/6). log55   
=(5/6).1  [log55=1]
=5/6


. এর মান নির্ণয় করঃ


) log5x=3

সমাধানঃ

log5x=3
বা,  x=53
বা,  x=125


) logx25=2

সমাধানঃ

logx25=2
বা,  25=x2
বা,  52=x2
বা,  5=x
বা,  x=5


) logx(1/16)=-2

সমাধানঃ

logx(1/16)=-2
বা,  1/16=x-2
বা,  ¼2=x-2
বা,  4-2=x-2
বা,  4=x
বা,  x=4


. দেখাও যে,


) 5log105-log1025=log10125

সমাধানঃ

LHS
=5log105-log1025
=5log105-log1052
=5log105-2log105
=3log105
=log1053
= log10125
=RHS (Proved)


) log10(50/147)=log102+2log105-log103-2log107

সমাধানঃ

LHS
= log10(50/147)
= log1050- log10147
= log10(255)- log10(377)
= log10(252)- log10(372)
=(log102+ log1052)-( log103+ log1072)
= log102+2 log105- log103-2 log107
=RHS (Proved)


) 3log102+2log103+log105=log10360

সমাধানঃ

LHS
= 3log102+2log103+log105
= log1023+ log1032+ log105
= log108+ log109+ log105
= log10(895)
= log10360
=RHS (Proved)


. সরল করঃ


) 7 log10(10/9)-2 log10(25/24)+3 log10(81/80)

সমাধানঃ

7 log10(10/9)-2 log10(25/24)+3 log10(81/80)
=7 log1010-7 log109-(2 log1025-2 log1024)+(3 log1081-3 log1080)
= log10107- log1097- log10252+ log10242+ log10813- log10803
= log10107+ log10242+ log10813-( log1097+ log10252+ log10803)
= log10 (107242813)- log10(97252803)
= log10(275732829393)- log10(97 525283103)
= log10(275732263636)- log10(314 5252295323)
= log10(21357314)- log10(314 57212)
= log10(21357314/314 57212)
= log102


) log7(5√7. √7)-log33√3+log42

সমাধানঃ

log7(5√7. √7)-log33√3+log42
= log771/5.71/2- log331/3+ log42
= log771/5+1/2- (1/3)log33+ log4√4
= log777/10-(1/3)log33+ log441/2
=(7/10) log77-(1/3)log33+(1/2) log44
=(7/10).1-(1/3).1+(1/2).1
=7/10-1/3+1/2
=(21-10+15)/30
=26/30
=13/15


) loge(a3b3/c3)+ loge(b3c3/d3)+ loge(c3d3/a3)-3 logeb2c

সমাধানঃ

loge(a3b3/c3)+ loge(b3c3/d3)+ loge(c3d3/a3)-3 logeb2c
= logea3b3- logec3+ logeb3c3- loged3+ logec3d3- logea3-loge(b2c)3
=( logea3b3+ logeb3c3+ logec3d3-( logec3-+loged3+ logea3+ logeb6c3)
=logea3b3b3c3c3d3-logec3d3a3b6c3
=logea3b6c6d3-logec6d3a3b6c3
=0


. x=2, y=3, z=5, w=7 হলে, নিচের প্রশ্নগুলো সমাধান কর।


) √(y3) এর 3 ভিত্তিক লগ নির্ণয় কর।

সমাধানঃ

log3√(y3)
=log3(y3)1/2
=log3y3/2
=log333/2
=(3/2).log33
=(3/2).1
=3/2


) w.log(xz/y2)-x.log(z2/x2y)+y.log(y4/x4z) এর মান নির্ণয় কর।

সমাধানঃ

w.log(xz/y2)-x.log(z2/x2y)+y.log(y4/x4z)
= 7.log(25/32)-2.log(52/223)+3.log(34/24 5)
=7 log(25)-7 log32-2 log52+ 2log(223)+3 log34-3 log(245)
=log(25)7-log(32)7-log(52)2+log(223)2+log(34)3-log(245)3
={log(25)7+ log(223)2+log(34)3 -{log(32)7+log(52)2 + log(245)3}
=log{(25)7(223)2(34)3-log(32)7(52)2(245)3
=log(21157314)-log(31457212)
=log(21157314)/(31457212)
=log2-1
=-1.log2
=-log2

) দেখাও যে,

 

 

log√y3+ylogx-(y/x)log(xz)

-----------------------------
log(xy)-logz


=


logy√y3

সমাধানঃ

LHS এর লব=
log√y3+ylogx-(y/x)log(xz)
= log33+3log2-(3/2).log(2.5)
= log33/2+log23-log(2.5)3/2
= log33/2+log(√4)3-log(2.5)3/2
= log33/2+log43/2-log(10)3/2
=(3/2).(log3+log4-log10)
=(3/2).log(3.4/10)
=(3/2).log(12/10)
=(3/2).log(6/5)

LHS এর হর=
=log(xy)-logz
=log2.3-log5
=log(2.3/5)
=log(6/5)

এখন, LHS=

 

 

(3/2).log(6/5)

-----------------
   log(6/5)


=


3/2
আবার,
RHS
= logy√y3
=log3√33
=log333/2
=(3/2). log33
=(3/2).1
=3/2

LHS=RHS (Proved)

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