$Y$ has the minimum Weibull $(\beta, \sigma)$ distribution if it has density function:
$$f_Y(y) = \frac{\beta}{\sigma} \left( \frac{y}{\sigma} \right)^{\beta -1} \exp \left[ - \left( \frac{y}{\sigma} \right)^\beta \right].$$
$X$ has the minimum Gumbel $(a,b)$ distribution if it has density function:
$$f_X(x) = \frac{1}{b} \exp \left[ \frac{x+a}{b} - \exp\left(\frac{x+a}{b} \right) \right] .$$
We also know that if $Y \sim \text{Weibull}(\beta,\sigma)$ then $X=\ln(Y) \sim \text{Gumbel}(a,b)$.
With this, I need to find a one-to-one relation between the parameters of the Weibull and the Gumbel distribution. With the change of variable formula, I have that the density of the logarithm of $Y$ is: $$ f_X(x)=\frac{\beta}{\sigma^\beta} \exp \left[ -\frac{\exp \left( x \beta \right)}{\sigma^\beta} + x \beta \right], $$ but I don't know what to do next.