Answer
$ 2x-z=2$
Work Step by Step
The vector equation is given by: $r(x,y,z)=r_0+t \nabla f(r_0)$
Given: $\ln(x^2+y^2)-z=0$
The equation of tangent line for $\nabla f(1,0,0)=\lt 2,0,-1 \gt$ is
$(2)(x-1)+(0)(y-0)-(1)(z-0)=0 \implies 2x-2-z=0$
Thus, $ 2x-z=2$
