How do you solve the equation e^2x - 2e^x - 3 = 0 ?

This is a 'quadratic in e^x'. The easiest way to recognise this when starting problems like this is to let y=e^x. As you practice problems like this, you will find yourself moving away from this step. But it is a useful illustration for the concept. Since y=e^x, y^2=e^2x. By substituting these facts into the equation, the equation becomes: y^2 - 2y - 3 = 0 This should now be in a more recognisable format, as quadratics like these will have come up in the Higher Specification for GCSE, or the C1 module in A-level. Assuming that this concept is well understood: This becomes: (y+1)(y-3)=0 Then: y = -1, or y = 3. Since y=e^x: e^x = -1, e^x = 3. By sketching a graph of e^x, it is evident that there are no solutions of e^x = -1. But by using logarithms, we can solve e^x = 3. This gives us our only solution: x = ln3.