Question

What is the functional equation of logarithm \[log(x.y)\] \[x,y > 0\] ? Conclude from the solution that it is \[\log(x^{n})=n.\log x, x > 0, n \in \mathbb{N}\]

Answer 1

i know functional equation of \[\ln(x.y)=\ln(x)+\ln(y)\] is but i dont know how to conclude \[\log(x^{n})=n.\log(x)\]

Answer 2

You know from the first property that
\[\ln(x^2)=\ln(x\cdot x)=\ln x+\ln x\]
You can extend this by induction:
\[\ln(x^n)=\ln\left(\prod_{k=1}^nx\right)=\sum_{k=1}^n\ln x=n\ln x\]

Answer 3

thanx super...