You have two curves, and . You wish to find the greatest number for which the curves intersect only once.
Let us substitute the second curve into the first one (since the values are presumed equal as there is intersection) :
Now let us reformulate the problem : you wish to find the greatest number for which this equation has only one solution for (equivalent to the statement "the curves only intersect once").
Expand this :
Ok, keep going :
Let's try to set this as a quadratic equation to use the discriminant :
Consider everything after the as the constant coefficient (yeah, who said we couldn't put 's in it ?)
You know that if the discriminant of a quadratic equation is equal to zero, then there is only one real solution. So, let us express the discriminant :
You want it to be equal to zero, so attempt to solve the following for (yes, now we care about because we are looking for the value of that makes the discriminant equal to zero) :
Cool, a straightforward quadratic equation ! Here we are using an advanced method : we are going to solve for , and this will serve as an intermediate result to determine which values of make the discriminant null. Get the discriminant first, for ease :
So if some makes the two curves intersect, then we have for the where the curves intersect (discard the negative value since a logarithm is always positive)
Now you can apply the same method but solving for instead, and then merge the results together to find a condition on allowing you to find the greatest that satisfies your problem.
PS : I think I got way too complicated here, I don't even know if this actually works, there surely is a simpler way to do this, what have you been doing last with your class ? (or elsewhere)