How to Interpret Lambert W Function? I just used an online calculator to calculate the following values of t.

1.) I am confused about how I am supposed to interpret the $W_{-1}$ and $W_0$. Could anyone help me out?
Here is my original equation:
$$1000e^{0.5t}=30000t$$
2.) How should I go about solving for t? I don't know Lambert W.
 A: 1.)
Lambert W isn't an elementary function. It's also not a function in the complex numbers. It's a relation in the complex numbers that consists of several function branches. Each of its branches is a function in the complex numbers.
The branches $W_{-1}$ and $W_0$ have real values for some of their real arguments. See the graphs of the real parts of $W_{-1}$ and $W_0$ at Wikipedia: Lambert W function.
If you need numerical values of LambertW, you need a calculator or software that has its series presentation implemented - like for all other functions and relations. Alternatively, you could use the equation $xe^x=-\frac{1}{60}$ and approximate the two x-values that satisfy this equation.
Since your online calculator found the solutions for $t$ in terms of Lambert W, it's also able to calculate the numerical values of these solutions.
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2.) Let's now calculate your solutions.
$$1000e^{0.5t}=30000t$$
The equation is an algebraic equation in dependence of the two monomials $t$ and $e^{0.5t}$. This kind of equations don't have solutions except $0$ that are elementary numbers.
Lambert W is the inverse relation of the function $x\mapsto xe^x$. We can calculate the inverse relation if we solve the equation $xe^x=y$ for $x$. We then have $x=\textrm{W}_k(y)\ \ (k\in\mathbb{Z})$. $k$ denotes the individual branches of Lambert W.
To solve equations of your kind, one has to rearrange the equation into the form $f(x)e^{f(x)}=c$, where $c$ is a constant, and apply Lambert W to the left-hand side and to the right-hand side of the equation.
$$1000e^{0.5t}=30000t$$
$$e^{0.5t}=30t$$
$$1=30te^{-0.5t}$$
$$30te^{-0.5t}=1$$
$$-0.5te^{-0.5t}=-\frac{1}{60}$$
$$-0.5t=\textrm{W}_k(-\frac{1}{60}),\ k\in\mathbb{Z}$$
$$t=-\frac{\textrm{W}_k(-\frac{1}{60})}{0.5},\ k\in\mathbb{Z}$$
For real values, we need $k=-1,0$.
