Yes, you can convert log base x to log base y. However, the point is that if OP is assuming that we're using log base 10, developing an intuitive understanding

Yes, you can convert log base x to log base y. However, the point is that if OP is assuming that we're using log base 10, developing an intuitive understanding of why a binary search is O(log n) is significantly harder than if OP knows that we're referring to log base 2.