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