Fractional part of $b \log a$ From the problem...

Find the minimal positive integer $b$ such that the first digits of $2^b$ are 2011

...I have been able to reduce the problem to the following instead: 

Find minimal $b$ such that $\log_{10} (2.011) \leq \operatorname{frac}(b~\log_{10} (2)) < \log_{10} (2.012)$, where $b$ is a positive integer

Is there an algorithm that can be applied to solve this or would you need to step through all possible b until you find the right solution?

You are looking for integers $b$ and $p$ such that $b\log_{10}2-\log_{10}(2.011)-p$ is small and positive. The general study of such things is called "inhomogeneous diophantine approximation," which search term should get you started, if you want something more analytical than a brute force search. As 6312 indicated, continued fractions come into it. 
