I have the joint pdf$$f(x_1,x_2)=x_1e^{-x_1(1+x_2)}I_{(0,\infty)}(x_1)I_{(0,\infty)}(x_2)$$and have to derive the joint pdf of $$Y_1=e^{-X_1}\qquad\text{ and }\quad Y_2=e^{-X_1X_2}$$ I set $x_1=-\ln(y_1)$ and $x_2=\ln(y_2)/\ln(y_1)$. When I plug these transforms into $f(x_1,x_2)$ and multiply with the absolute determinant of the Jacobian $|\det(J)|=1/(y_1y_2\ln(y_2))$, I get a negative result. Did I make a mistake?

I have the joint pdf$$f(x_1,x_2)=x_1e^{-x_1(1+x_2)}I_{(0,\infty)}(x_1)I_{(0,\infty)}(x_2)$$and have to derive the joint pdf of $$Y_1=e^{-X_1}\qquad\text{ and }\quad Y_2=e^{-X_1X_2}$$ I set $x_1=-\ln(y_1)$ and $x_2=\ln(y_2)/\ln(y_1)$. When I plug these transforms into $f(x_1,x_2)$ and multiply with the absolute determinant of the Jacobian $|\det(J)|=1/(y_1y_2\ln(y_2))$, I get a negative result. Where did I make a mistake?

I have the pdf $f(x_1)=x_1e^{-x_1(1+x_2)}I_{(0,\infty)}(x_1)$ and have to derive the joint pdf of $Y_1=e^{-X_1}$ and $Y_2=e^{-X_1X_2}$. I get $x_1=-\ln(y_1)$ and $x_2=\ln(y_2)/\ln(y_1)$. When I plug these into $f(x_1,x_2)$ and multiply with the absolute determinant of the Jacobian $|\det(J)|=1/(y_1y_2\ln(y_2))$, I get a negative result. Did I make a mistake or is there an error in the exercise?

I have the joint pdf$$f(x_1,x_2)=x_1e^{-x_1(1+x_2)}I_{(0,\infty)}(x_1)I_{(0,\infty)}(x_2)$$and have to derive the joint pdf of $$Y_1=e^{-X_1}\qquad\text{ and }\quad Y_2=e^{-X_1X_2}$$ I set $x_1=-\ln(y_1)$ and $x_2=\ln(y_2)/\ln(y_1)$. When I plug these transforms into $f(x_1,x_2)$ and multiply with the absolute determinant of the Jacobian $|\det(J)|=1/(y_1y_2\ln(y_2))$, I get a negative result. Did I make a mistake?