I would appreciate it if someone let me know when it's appropriate to give logarithmic expressions a common denominator. For example, when finding the derivative of $f(x)= x^{\log_2 x}$ using logarithmic differentiation, before multiplying by y, I have $\frac{\ln x}{x \ln 2}$ + $\frac{\log_2 x}{x}$. At this point, my book simplifies by giving both expressions a common denominator of $x ln 2$ and ends up with $f'(x)= \frac{2 \ln x}{x \ln 2}$*$x^{\log_2 x}$ ($x^{\log_2 x}$ was the $y$).
The second example is $f(x)= 2^x*\log_3 7^{x^2-4}$. Not applying logarithmic differentiation, the derivative is $f'(x)= (2^x\ln2)(\log_3 7^{x^2-4}) + \frac{(2x) \ln7}{\ln 3}*2^x$. Why wasn't it necessary to give both expressions a common denominator? I just find it confusing knowing what to do when it comes to simplifying logarithmic expressions. Thanks for any answers!