Unbiased estimator for $\lambda$

We were asked to find the UMVUE for $$\lambda$$ using the Lehmann-Scheffe Theorem, given a random sample from $$f_X(x;\lambda)=\frac{1}{\lambda}x^{\frac{1}{\lambda}-1}, 0 0$$.

I know that in order to use the theorem, I need to find an unbiased estimator, and a complete sufficient statistic for $$\lambda$$. I already found a c.s.s. but I am quite lost on how to find the unbiased estimator.

Let $$X_1,\dots,X_n$$ be a random sample from a $$\text{Beta}(1/\lambda,1)$$ distribution. Hence, $$X_i$$ has the desired density: $$f(x_i\mid\lambda)=(1/\lambda)\,x_i^{(1/\lambda)-1}$$, for $$0, and $$\lambda>0$$. It's not difficult to show that $$\mathbb{E}[\log X_i]=\psi(1/\lambda)-\psi(1+1/\lambda)=\lambda$$, in which $$\psi$$ is the digamma function. It follows that $$\hat{\lambda}=(1/n)\sum_{i=1}^n \log X_i$$ is an unbiased estimator of $$\lambda$$. By the way, this is the maximum likelihood estimator of $$\lambda$$, as you can easily verify.