Solve 2 ln (√x) - ln (1 - x ) = 2 .
Solution: Firstly get the two logarithms combined in a single logarithm.
2 ln (√x) - ln (x - l) = 2
ln ((√x)2 ) ln (1 - x ) = 2
ln ( x ) - ln (1 - x ) = 2
ln(x/(1-x)=2
Now, exponentiate both sides & solve out for x.
x /1 - x=e2
x= e2
x = e2 (1 - x )
x = e2 - e2 x
x (1 + e2 ) = e2
x = e2/1 + e2 =0.8807970780
At last, we just need to ensure that the solution, x = 0.8807970780 , doesn't generate negative numbers in both of the original logarithms. This doesn't, so it is in fact our solution to this problem.
While solving equations with logarithms it is considerable to check your potential solutions to ensure that they don't produce logarithms of negative numbers or zero. It is also significant to ensure that you do the checks in the original equation. If you verify them in the second logarithm above (after we've combined the two logs) both solutions will seem to work! It is because in combining the two logarithms we've in fact changed the problem. Actually it is this change that introduces the extra solution which we couldn't use!
Also be careful in solving equations of logarithms to not get locked into the idea that you will get two potential solutions & only one of these will work. This is possible to have problems where both are solutions & where neither are solutions.