Before we apply intermediate value theorem, let’s discuss why the intersection can only happen in rather than .
We know from the question that when . Since the exponential function is an increasing function, we have when . On the other hand, is decreasing, and hence so is . This follows that when . This implies that for , we have
.
(Note that .) In other words, for any point in .
In a similar manner, we can show that when , we have that
i.e.,
on
The above analysis would be more obvious if you draw the graphs of and .
Now, we find the value which makes have at least one solution on the interval , by using the intermediate value theorem.
On the interval , let
(Here, we use on .)
It is time to apply intermediate value theorem, note that is continuous in the interval , and
To make have at least a zero in , by the theorem, we need to make sure that and have opposite signs. i.e., .
Solving this inequality gives us that
Thus, we have at least one solution to (i.e., ) if