disk of convergence of composition of p-adic functions

I watched a video that sketched out the regions of convergence for both the p-adic logarithm and the p-adic exponential functions. I thought about all this and asked myself:

How would I find the region of convergence of $f(x)=\exp\bigg( \frac{1}{\log_p x} \bigg)$?

Well I know that the disk of convergence for $\log_p (x)$ is $D(1,1^-)=\big\lbrace x\in \Bbb C_p: |x|_p<1 \big\rbrace$, and I also know that the disk of convergence for $\exp$ is $D(1,r_p^-)=\big\lbrace x\in \Bbb C_p : |x|_p<r_p \big\rbrace $ where $r_p=|p|^{\frac{1}{p-1}}.$

So would the disk of convergence of $f$ be $D= \big\lbrace x\in \Bbb C_p: |x|_p<\exp\bigg(\frac{p-1}{\log_p |p| }\bigg)\big\rbrace?$