How to find the big O notation of $\log^b x$
How would you determine big o notation for this? I don't think you can simply do O(log(x)^b) can you?


$\mathcal O\left(\log^b x\right)$ or $\mathcal O\left(\left(\log x\right)^b\right)$ is correct.


  x^3 + log(x)^b


Assuming $b$ is a constant. You always take the fastest growing term in a polynomial. $T(x)=\mathcal O\left(\log^b x\right)$ is called polylogarithmic time. In this case, $\mathcal O\left(x^3\right)$ grows faster than $\mathcal O\left(\log^b x\right)$.

You can see a list of different complexities (sorted from lowest to highest) here.