Find lower bound of function ？

Question

Can someone help me find a lower bound to the function

$$\ln(1 - 0.5^u), u\ge0$$

I tried to use the Taylor formula, but it didn't seem to work. I would be grateful if anyone could give me some advice.

Answer 1

As $u\rightarrow 0$, the expression tends to $-\infty$, so there is no lower bound.

Answer 2

As a general approach you could try to find a minimum point.

$$y=\ln(1-0.5^u).$$

Differentiating,

$$y'=\frac{\log(2)0.5^u}{1-u^{0.5}}.$$

To find stationary points, set equal to zero to obtain

$$0.5^u=0,$$

which has no solutions, so there are no stationary points. Hence, no minimum and so no lower bound.