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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=5243=35
    Generalising ab=c(Exponential form)(logarithmic form)logac=bc=ab(log of c to base a is b)
    Eg: 23=18=0.125log20.125=3
    |||ly(81)12=9log819=121)x0=1logx1=0 Eg:log51=0   log10=0x1=xlogxx=1 Eg:log55=1   log1010=1
    Ques: log101000=3,   log1011000=3
    log2(x24)=5x=±6,logx64=32x=16log7(2x21)=2x=±5
    Laws of logarithms
    1) Product law loga(m×n)=logam+loganloga(m×n)=k(m×n)=akpm cal     qqn=aRHS=P+qloga(m×n)=loga(ap×aq)=loga(ap+q)=km×n=m+m+ntimes    xap+q=akk=p+qloga(ap+q)=(p+q)
    LHS

    2) Quotient law loga(mn)=logamlogan=pq
    logamloganmlog a

    3) Power law loga(m)n=n logam
    LHS
    logamn=logam×m×mn times=logam+logan+log mn times=n×logamlogam=pm=aplogan=qn=aq∣ ∣let, loga(mn)=Kmn=akapaq=apq=akk=pqloga(mn)=pq=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 b2 log c (logarithmic expansion)
    Ques: 2log10  x+1=log10250  x=?
    log10x2=log1025x2=25x=±5=5
    log105=x5=10xlog105=x5=(10)2

    Solve for xlogx49logx7+logx13×3+2=0logx49logx7logx343+logxx2=0log749=x49=(3)2
    logab=1logba 

    logab=K 
    b=ak    

    1logba=y
    logb               
     a=1y
    a=b1y=axy
    x=kyk=y

    Ques log(ab2)=12(loga+logb)
    a2+b2=Kab Find K
    ab2=2aba2+b22ab4ab=0a2+b2=6abk=6
    Indices
    The product of m factors each equal to 'a' is represented by a^m
    So, am=a.a.am times
    Here, a = base, m = index
    Law of indices 1) am+n=am.an  2)am=1am  a03) a0=1 a0  4) amn=aman a05) (am)n=amn  6) apq=qap  7) (ab)m=am.bm
    logba=logmalogmb
    alogaN=N=Nlogaa=Na1
    Proof:  logaN=k
    loga(alogaN)=logaKlogaN1=logaKN=K

    Antilog
    If log2572=929=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: log55555=x  n=?(1)
    55555=y5y=y5y=y2log55=1y25y=0y(y1)=0y=0,5

     

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