Power Law

${\log }_b(x^n)=n{\log }_{b}x$ We are proving $y={\log }_b(x)$ Bring the moon(the exponent) down to earth(regular coefficient)

Proof

${\log }_b(x^n)=?$

1. Write the equation in exponential form $b^y=x$
2. Raise both sides to the exponent n $b^{y^n}=x^n$
3. Apply the power law of exponents $b^{yn}=x^n$
4. Rewrite in the logarithmic form ${\log }_b(x^n)=ny$
5. Substitute $y={\log }_{b}x$ to eliminate y $n{\log }_b=x$ We used to have an exponent. But it is gone.