Logarithm

The logarithm of a number to a given base is the index of the power to which the base must be raised to be equal to that number.

Thus if a^x= N, then x is the logarithm of N to the base a, and written as x =log _aN

For example, 100 =10²                  so     log₁₀¹⁰⁰ = 2

 and (.0001) =(.1)⁴                        so     log (.0001) = 4

Conversely, if  log_aN = x           then             N = a^x.

Some important deductions.

(i)                         log_a¹ = 0 or log of 1 to any base = 0 [ a⁰=1]

(ii)                      log_aa = 1 or log of any number to the same 

           base =1 [ a’ =a]

(iii)                   log_a∞= ∞, for values of a>1.   [ a^∞ =∞]

(iv)                   log_a⁰= -∞, for values of a>1.   

                [ a^-∞ =1/a^∞= 1/^∞=0]