Exponential equation, variable on both sides

Question

I need help with solving the following equation for t

$Re^{vht}=v_1t$ where $R,v,h,v_1$ are constants. Any suggestions appreciated.

Answer 1

Rewrite this as $$ -\frac{Rvh}{v_1} = -vht\cdot e^{-vht} $$ Applying the Lambert $W$ function we get $$ W\left(-\frac{Rvh}{v_1}\right) = -vht\\ t = -\frac{W\left(-\frac{Rvh}{v_1}\right)}{vh} $$ Using the $W$ function isn't much more than a rewriting, but there is nothing else that can be done when we have $t$ both in the exponent and outside it.

Answer 2

$\textbf{Note this is beyond pre-calc}$

Apart from the Lambert Function which @Arthur has mentioned, lets play with some cases where we can assume $vht<<1$ then we can approximate our equation as $$ R(1+vht) = v_1t \implies t = \frac{R}{v_1-vhR} $$ we obviously require $v_1 > vhR$.

But beyond this you would have to apply a numerical scheme to solve this $$ R\mathrm{e}^{vht} - v_1t = f(t) = 0 $$ which is a classic root-finding problem.