Solution
= Solution
If tangency occurs at $x_*$, the <slope> must satisfy both $k=2e^{2x_*}$ and $kx_*=e^{2x_*}$. Eliminating $k$ gives $2x_*=1$, so the <tangent through the origin to an exponential curve> has
$$
\boxed{k=2e.}
$$
= Solution
If tangency occurs at $x_*$, the <slope> must satisfy both $k=2e^{2x_*}$ and $kx_*=e^{2x_*}$. Eliminating $k$ gives $2x_*=1$, so the <tangent through the origin to an exponential curve> has
$$
\boxed{k=2e.}
$$