How can I go about getting the inverse of$ f(t) = e^{-.001t}\cdot e^{-.005t}$? I have found a couple of calculators online that suggest that the answer is: $t=-166.667\ln(y)$, but I would like to know the steps. I know how to solve this for a single Euler term.
2026-04-05 23:15:41.1775430941
Inverse of function with two Exponential Eulers Terms
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$f(t) = e^{-0.001t}\cdot{e^{-0.005t}}\\ f(t) = e^{-0.001t+(-0.005t)}\\ f(t) = e^{-0.006t} \\ y = e^{-0.006t} \\ inverse:t = e^{-0.006y}\\ -0.006y = ln(t) \\ y = -\frac{1}{0.006}\ln(t)\\ y = -166.667\ln(t)$
EDIT: Note that $ln{a^{b}} = b\ln{a}$ and $\ln{e} = 1$