Question

prove that logaXn=nlogaX where a, x and n are positive real numbers and a is not equal to 1​

Answer 1

Answer:

It is only true when xn>0, so we assume it.

We'll use the following definition, which is how Wikipedia and Wolfram define it:

logbx=k⟺bk=x

together with the exponentiation rule: bxy=(by)x

logb(xn)=nlogb|x|⟺bnlogb|x|=xn

⟺(blogb|x|)n=xn⟺|x|n=xn

⟺|xn|=xn,

which is true.