Find a logarithmic tangent and a two-intersection slope range

The curve $C$ has equation $2^{-y}=x$. Its point $A$ has x-coordinate $a>0$. Show that $dy/dx=-1/(a\ln2)$ at $A$, and find the equation of the tangent there.

Hence find the tangent to $C$ that passes through the origin. The line $y=mx$ intersects $C$ at two distinct points. Write down the range of $m$.