expressing function with series

I got a function to express v*t=(S₀-S_t)+K_m*ln(S₀/S_t). The only unknown here is S_t. I solve it for S_t and get

S_t=LambertW((S₀*e)/K_m)*K_m)

And I need to express this in rows then t=0..x (t=time) I know I should use series or asympt because t is variable, but I keep getting errors. Is there any way to express this function in rows??and 2nd question would be can I solve the function to find S_t without that LambertW. If so how do I do it.