untitled

Now,MN axay axy
?loga(MN) xy,orloga(MN) logaMlogaN [putting the values of x,y]

?loga(MN) logaMlogaN. (proved)

Note 1.loga(MNP.....) logaMlogaNlogaP.........

Note 2.loga(MrN)zlogaMrlogaN

Formula 3. M N
N

M loga loga loga

Proof : Let logaM x,logaN y;

?M ax,N ay

Now, y xy

x a a

a N

M

? x y
N

M a ̧^ ¹

· ̈ ©

log§

? M N
N

M loga ̧ loga loga ¹

· ̈ ©

§ (proved).

Formula 4.logaMr rlogaM.

Proof : : Let logaM x;?M ax

? (M)r (ax)r; orMr arx ? logaMr rx; orlogaMr rlogaM ? logaMr rlogaM. (proved ).

N.B. : (logaM)rzrlogaM

Formula 5. loga M = logb M u loga b, (change of base)
Proof : Let, logaM x,logbM y

? ax M,by M ? ax by, or ax y by y

1 1
( ) ( )
or y

x b a

? log b, y

x

(^) a or x ylogab
or , x ylogab, or logaM logbMulogab (proved).
Corollary : ,
log
1
log
a
b
b
a^ or,
b
a
a
b
log
1
log