New logarithms identity?

Question

$$\log_{x^{\frac{1}{n}}} x = \frac{\log_x x}{\log_x x^{\frac1 n}} = \frac{\log_x x}{\frac1 n} = n \log_x x = \log_xx^n$$

Hence, can we say that $\log_{x^{\frac{1}{n}}} x = \log_xx^n$, or is this identity not proven?

Answer 1

Using $\log_ax=\frac1{\log_xa}$,
$$\log_{x^{(\frac1n)}}x=\frac1{\log_xx^{(\frac1n)}}=\frac1{\frac1n}=n$$
So this is just a manipulation, not an identity.

Answer 2

As already stated in the comments, we have $$\log_{x^{\frac{1}{n}}} x=\frac{\log x}{\log x^{\frac{1}{n}}}=\frac{\log x}{\frac1n\log x}=n\frac{\log x}{\log x}=n$$ Would I call this an identity? Probably not. Its more of a simplification. Study the proofs of some already existing log identities for some inspiration.