3. The graphs of the exponential function $y=a^{x}$ and the logarithmic function $y=\log _{a} x$ (where $a>0, a \neq 1$) are denoted as $C_{1}$ and $C_{2}$, respectively. Point $M$ is on curve $C_{1}$, and line segment $O M$ (where $O$ is the origin) intersects curve $C_{1}$ at another point $N$. If there exists a point $P$ on curve $C_{2}$, and the x-coordinate of point $P$ is equal to the y-coordinate of point $M$, and the y-coordinate of point $P$ is twice the x-coordinate of point $N$, then the coordinates of point $P$ are ( ).
(A) $\left(\log _{a} 4,4\right)$
(B) $\left(4, \log _{a} 4\right)$
(C) $\left(2, \log _{a} 2\right)$
(D) $(4, \log 2)$
3. B.

Let $M\left(x_{1}, a^{x_{1}}\right), N\left(x_{2}, a^{x_{2}}\right)$, then we have $x_{p}=a^{x_{1}}, y_{p}=2 x_{2}=\log _{a} a^{x_{1}}=x_{1}$. Given that points $O, N, M$ are collinear, hence $\frac{x_{1}}{x_{2}}=\frac{a^{x_{1}}}{a^{x_{2}}}=2$,
which means $a^{2 x_{2}}=2 a^{x_{2}}$.
Solving this, we get $x_{2}=\log _{a} 2, x_{1}=\log _{a} 4$. Therefore, $P\left(4, \log _{4} 4\right)$.