Definition of Logarithm of Positive Number with Given Base

Logarithm of positive number `x` with base `a` (`a>0,a!=1`) is called such exponent to which we need to raise number `a` to obtain `x`: `color(blue)(a^(log_a(x))=x)`.

Equality `log_a(x)=y` means that `a^y=x`.

For example, `log_3(81)=4` because `3^4=81`; `log_10(0.001)=-3` because `10^(-3)=0.001`; `log_(1/2)(sqrt(2))=-1/2` because `(1/2)^(-1/2)=2^(1/2)=sqrt(2)`.

In record `log_a(x)` number `a` is called base of logarithm.

From definition of logaritm it follows that:

  • `log_a(1)=0` because `a^0=1`.
  • `log_a(a)=1` because `a^1=a`.
  • In general, `log_a(a^r)=r` because `a^r=a^r`.