Rate of growth of bacteria is proportional to the number of bacteria present at time. If $x$ is the number of bacteria present at any instant $t$, then which one of the following is correct? (Let the proportional constant equal to $2$)

The rate of growth is given by $\frac{dx}{dt}$.According to the problem:
$$\frac{dx}{dt}\propto x⟹\frac{dx}{dt}=kx$$
Given the proportionality constant $k=2$:
$$\frac{dx}{dt}=2x$$
Separate the variables:
$$\frac{dx}{x}=2dt$$
Integrate both sides:
$$\int \frac{1}{x}dx=\int 2dt$$
$$\ln x=2t+C_1$$
Exponentiate both sides:
$$x=e^{2t+C_1}=e^{C_1}\cdot e^{2t}$$
Let $C=e^{C_1}$ be the arbitrary constant:
$$x=C{e}^{2t}$$
(Note: The solution $x=C{e}^t$ in the source seems to have a typo regarding the exponent; based on $k=2$, it must be $2t$).