Let us show that the logarithm has the property that given an arbitrary real number , there is always a sufficiently large value of the integer above which . We simply promote to a continuous real variable and consider the function
Is is quite clear that this function diverges to positive infinity as . If we calculate the first and second derivatives of we get
It is now clear that there is a single critical point where the first derivative is zero, that is where , given by . At this point we have for the second derivative
which is positive, implying that the critical point is a local minimum. Since there is no other minimum, maximum or inflection point, it becomes clear that the function must decrease from positive infinity as increases from zero, go through the point of minimum at , and then increase without limit as . At this point of minimum we have for the function itself,
It follows that, if , then we must have for all . This corresponds to . On the other hand, if , then there are two solutions and of the equation , that coincide if . This corresponds to . In this case the function is positive within the interval , negative within and positive for .
Therefore, for all possible values of there is a value of , either or , such that for the function is positive, and therefore
Translating the statement back in terms of , we have that given any real number , there is a minimum value of the integer such that . Thus the required statement is established.