Enzymatic action may be described as follows:
$$\ce{Enzyme + Substrate <=>[k_1] ES complex ->[k_\mathrm{2}] Enzyme + Product}$$
The initial rate of enzyme-catalyzed reactions can be described by the Michaelis-Menten equation:
$$\mathrm{rate} = \frac{V_\mathrm{max}[\ce{S}]}{K_\mathrm{M} + [\ce{S}]} = \frac{k_\mathrm{cat}[\ce{E}][\ce{S}]}{K_\mathrm{M} + [\ce{S}]}$$
where $V_\mathrm{max}$ is the maximum rate, $[\ce{S}]$ the substrate concentration, $[\ce{E}]$ is the enzyme concentration, $K_\mathrm{M}$ is the Michaelis constant and $k_\mathrm{cat}$ is the number of catalytic cycles per second.
It is known that
$$V_\mathrm{max} = k_2[E]_0$$
and by inspecting the equation above we can deduce that
\begin{align} V_\mathrm{max} &= k_\mathrm{cat}[\ce{E}]\\ \implies k_2[\ce{E}]_0 &= k_\mathrm{cat}[\ce{E}] \end{align}
How, and why, does this equality hold?
How do we know when to use ${[\ce{E}]_0}$ and $[\ce{E}]$ in rate equations, and what are the implications of using either?
${[\ce{E}]_0}$ refers to the enzyme concentration at the start of the reaction, and [E] refers to the concentration of enzyme at any point in time during the course of the reaction.