Question 115743

All real logarithms below have positive arguments and a positive base different from 1. In the power law, the argument before raising it to a power is positive. In change of base, the new base must also be positive and different from 1. The reciprocal identity requires both bases to differ from 1: \(x,y>0\), \(x,y\ne1\).

Change of Base:

\({{\log }_{a}}b=\frac{{{\log }_{c}}b}{{{\log }_{c}}a}\)

e.g. \({{\log }_{2}}5=\frac{{{\log }_{7}}5}{{{\log }_{7}}2}\) , \({{\log }_{11}}2=\frac{{{\log }_{p}}2}{{{\log }_{p}}11}\)

Important notes:

\({{\log }_{x}}y=\frac{1}{{{\log }_{y}}x}\)