I understand that when using l'hopital rule to find the limits of a fuction with power unknown( eg X^x), we have to use natural logarithm. But can we use logarithm to base 10 instead?
In mathematics there really is no such thing as a logarithm with base 10. Because there is nothing important about the number 10, except that it is a number of the decimal system we use, and might be convient for certain exponential calculations. You really only use that logarithm in high school to get introduced to logarithms. Mathematicians always use the natural logarithm. They do not even call it the 'natural' logarithm - they simply say the logarithm function. Because for them there is only one such function. And when they write they mean , that distinction is again something you see in high-school but stop seeing when you do more advanced math.