Add assumptions to Solve command

Question

I want Mathematica to solve the following by assuming the part within the Log function, which is ((P - T) (L - P0))/((L - P) (P0 - T)), to be positive and give a result that is not a conditional expression. How do I add assumptions to the Solve command?

Clear[k, P, t, T, L, P0]
Solve[L/(L - T) (Log[((P - T) (L - P0))/((L - P) (P0 - T))]) == k*t,
P]

Any suggestion is much appreciated. Thanks

Answer 1

The ConditionalExpression output by Solve in your case does not really depend on the whole argument of the Log function in your original equation, but only on the following expression:

-π < Im[(k t (L - T))/L] ≤ π

If it is possible in your case to make assumptions on the values of those parameters, you could then try to Simplify the output of Solve using the Assuming function.

For instance, if your parameters are real numbers, and L is positive, you could write:

Assuming[
  { {k, t, L, T, P} \[Element] Reals, L > 0},
  Simplify[
    Solve[L/(L - T) (Log[((P - T) (L - P0))/((L - P) (P0 - T))]) == k*t, P]
  ]
]

The result of that last command no longer depends on a ConditionalExpression.