Logarithms

The log of a positive quantity N to a given base a is defined as the index of the power to which the base a must be raised to make it equal to the given quantity N. Thus:

ax = N  &  logaN = x  

So

Consider

 x= 0  then a0 = 1 so N = 1

So        

loga1 = 0

Consider

x = 1 then a1 = a so N = a

So

logaa = 1

Multiplication and Division

         &         

So

        

So

loga(NM) = logaN + logaM

Also

        

So

loga(N/M) = logaN - logaM


Converting bases

Let         

x = logbN                …..(1)

∴        bx = N

Take loga of both sides

loga(b)x = logaN

xlogab = logaN

Using (1)

logbN.logab = logaN