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

  লগারিদমঃ সরল, প্রমান ও সমাধান

১. মান নির্ণয় করঃ

ক) log₃81

সমাধানঃ

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

খ) log₅³√5

সমাধানঃ

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

গ) log₄2

সমাধানঃ

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

ঘ) log_2√5400

সমাধানঃ

log_2√5400
= log_2√516✕25
= log_2√52⁴✕5²
= log_2√52⁴✕(√5)⁴
= log_2√5(2√5)⁴
=4. log_2√52√5 [log_2√52√5=1]
=4.1
=4

ঙ) log₅(³√5. √5)

সমাধানঃ

log₅(³√5. √5)
= log₅5^1/3.5^1/2
= log₅5^1/3+1/2
= log₅5^1/3+1/2
= log₅5^5/6
=(5/6). log₅5   
=(5/6).1  [log₅5=1]
=5/6

২. এর মান নির্ণয় করঃ

ক) log₅x=3

সমাধানঃ

log₅x=3
বা,  x=5³
বা,  x=125

খ) log_x25=2

সমাধানঃ

log_x25=2
বা,  25=x²
বা,  5²=x²
বা,  5=x
বা,  x=5

গ) log_x(1/16)=-2

সমাধানঃ

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

৩. দেখাও যে,

ক) 5log₁₀5-log₁₀25=log₁₀125

সমাধানঃ

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

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

সমাধানঃ

LHS
= log₁₀(50/147)
= log₁₀50- log₁₀147
= log₁₀(2✕5✕5)- log₁₀(3✕7✕7)
= log₁₀(2✕5²)- log₁₀(3✕7²)
=(log₁₀2+ log₁₀5²)-( log₁₀3+ log₁₀7²)
= log₁₀2+2 log₁₀5- log₁₀3-2 log₁₀7
=RHS (Proved)

গ) 3log₁₀2+2log₁₀3+log₁₀5=log₁₀360

সমাধানঃ

LHS
= 3log₁₀2+2log₁₀3+log₁₀5
= log₁₀2³+ log₁₀3²+ log₁₀5
= log₁₀8+ log₁₀9+ log₁₀5
= log₁₀(8✕9✕5)
= log₁₀360
=RHS (Proved)

৪. সরল করঃ

ক) 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 (10⁷✕24²✕81³)- log₁₀(9⁷✕25²✕80³)
= log₁₀(2⁷✕5⁷✕3²✕8²✕9³✕9³)- log₁₀(9⁷✕ 5²✕5²✕8³✕10³)
= log₁₀(2⁷✕5⁷✕3²✕2⁶✕3⁶✕3⁶)- log₁₀(3¹⁴✕ 5²✕5²✕2⁹✕5³✕2³)
= log₁₀(2¹³✕5⁷✕3¹⁴)- log₁₀(3¹⁴✕ 5⁷✕2¹²)
= log₁₀(2¹³✕5⁷✕3¹⁴/3¹⁴✕ 5⁷✕2¹²)
= log₁₀2

খ) log₇(⁵√7. √7)-log₃³√3+log₄2

সমাধানঃ

log₇(⁵√7. √7)-log₃³√3+log₄2
= log₇7^1/5.7^1/2- log₃3^1/3+ log₄2
= log₇7^1/5+1/2- (1/3)log₃3+ log₄√4
= log₇7^7/10-(1/3)log₃3+ log₄4^1/2
=(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

গ) log_e(a³b³/c³)+ log_e(b³c³/d³)+ log_e(c³d³/a³)-3 log_eb²c

সমাধানঃ

log_e(a³b³/c³)+ log_e(b³c³/d³)+ log_e(c³d³/a³)-3 log_eb²c
= log_ea³b³- log_ec³+ log_eb³c³- log_ed³+ log_ec³d³- log_ea³-log_e(b²c)³
=( log_ea³b³+ log_eb³c³+ log_ec³d³-( log_ec³-+log_ed³+ log_ea³+ log_eb⁶c³)
=log_ea³b³✕b³c³✕c³d³-log_ec³✕d³✕a³✕b⁶c³
=log_ea³b⁶c⁶d³-log_ec⁶d³a³b⁶c³
=0

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

ক) √(y³) এর 3 ভিত্তিক লগ নির্ণয় কর।

সমাধানঃ

log₃√(y³)
=log₃(y³)^1/2
=log₃y^3/2
=log₃3^3/2
=(3/2).log₃3
=(3/2).1
=3/2

খ) 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(2✕5/3²)-2.log(5²/2²✕3)+3.log(3⁴/2⁴ ✕5)
=7 log(2✕5)-7 log3²-2 log5²+ 2log(2²✕3)+3 log3⁴-3 log(2⁴✕5)
=log(2✕5)⁷-log(3²)⁷-log(5²)²+log(2²✕3)²+log(3⁴)³-log(2⁴✕5)³
={log(2✕5)⁷+ log(2²✕3)²+log(3⁴)³ -{log(3²)⁷+log(5²)² + log(2⁴✕5)³}
=log{(2✕5)⁷✕(2²✕3)²✕(3⁴)³✕-log(3²)⁷✕(5²)²✕(2⁴✕5)³
=log(2¹¹✕5⁷✕3¹⁴)-log(3¹⁴✕5⁷✕2¹²)
=log(2¹¹✕5⁷✕3¹⁴)/(3¹⁴✕5⁷✕2¹²)
=log2^-1
=-1.log2
=-log2

গ) 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)
= log√3³+3log2-(3/2).log(2.5)
= log3^3/2+log2³-log(2.5)^3/2
= log3^3/2+log(√4)³-log(2.5)^3/2
= log3^3/2+log4^3/2-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
= log_y√y³
=log₃√3³
=log₃3^3/2
=(3/2). log₃3
=(3/2).1
=3/2

∴LHS=RHS (Proved)