Question

# Give a brief description about logarithm.

Solution

## The correct option is Logarithms are used to make the long and complicated calculations easy. Eg: 35=243→ We have a relation between 3, 5 and 243 Now, same relation between 3,5 and 243 can be written as log3243=5⇒243=35 Generalising ab=c(Exponential form)⇒(logarithmic form)logac=b⇒c=ab(log of c to base a is b) Eg: 2−3=18=0.125→log20.125=−3 |||ly(81)12=9⇒log819=121)x0=1⇒logx1=0 Eg:log51=0   log10=0x1=x⇒logxx=1 Eg:log55=1   log1010=1 Ques: log101000=3,   log1011000=−3 log2(x2−4)=5x=±6,logx64=32x=16log7(2x2−1)=2x=±5 Laws of logarithms 1) Product law →loga(m×n)=logam+loganloga(m×n)=k⇒(m×n)=akpm cal     qqn=a∴RHS=P+qloga(m×n)=loga(ap×aq)=loga(ap+q)=km×n=m+m+……ntimes    xap+q=akk=p+q∴loga(ap+q)=(p+q) LHS 2) Quotient law loga(mn)=logam−logan=p−q logamlogan≠mlog a 3) Power law loga(m)n=n logam LHS logamn=logam×m×m……n times=logam+logan+log m……n times=n×logamlogam=p⇒m=aplogan=q⇒n=aq∣∣ ∣∣let, loga(mn)=Kmn=ak⇒apaq=ap−q=ak∴k=p−q∴loga(mn)=p−q=RHS Logarithms to the base 10 are called common logarithms If no base is given - take it as 10 Expansion of expression using logarithms y=a5×b3e2 logy=5 log a+3 log b−2 log c (logarithmic expansion) Ques: →2log10  x+1=log10250  x=? ⇒log10x2=log1025∴x2=25x=±5=5 log10−5=x−5=10xlog−10−5=x−5=(−10)2 Solve for x→logx49−logx7+logx13×3+2=0logx49−logx7−logx343+logxx2=0⇒log−749=x49=(−3)2 logab=1logba  logab=K  b=ak     1logba=y ⇒logb                 a=1y a=b−1y=axy ∴x=ky⇒k=y Ques →log(a−b2)=12(loga+logb) a2+b2=Kab Find K a−b2=2√ab⇒a2+b2−2ab−4ab=0a2+b2=6abk=6 Indices The product of m factors each equal to 'a' is represented by a^m So, am=a.a.a……m times Here, a = base, m = index Law of indices →1) am+n=am.an  2)a−m=1am  a≠03) a0=1 a≠0  4) amn=aman a≠05) (am)n=amn  6) apq=q√ap  7) (ab)m=am.bm logba=logmalogmb alogaN=N=Nlogaa=Na≠1 Proof:  logaN=k loga(alogaN)=logaKlogaN−1=logaK∴N=K Antilog If log2572=9→29=572 Antilog29=572Antilog29=29 logN1N=−1     logbma=1mlogba Whomever the no. and base are on the same side of unity then logarithm of that no. to the base is +ve When no. and base are on different sides of unity then logarithm of that no. to the base is -ve →log10100=2Butlog110100=−2 Ques: log5√5√5√5√5……=x  n=?→(1) √5√5√5√5√5……=y∴√5y=y→5y=y2∴log55=1⇒y2−5y=0y(y1)=0y=0,5

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