My strategy for evaluating the logarithms of surds that are in the form of: log$_b$ ($^x\surd$rad) = y involves transforming the surd to exponent form, where the radicand becomes the base and the index is the reciprocal of the surds’ and putting the whole thing into what I think of as index notation (b$^x$= y)
For example, evaluating log$_3$ ($^4\surd$9) = x converts to log$_3$ (9$^{1/4}$) = x
In the next stage of the process, I rearrange the initial statement to the form b$^x$ = y and perform further operations on the Left Hand Side (LHS) to deduce x.
For example:
step 0: 3$^x$ = 9$^{1/4}$
step 1: (3$^2$)$^x$ = 9$^{1/4}$
step 2: (3$^2$)$^{1/4}$ = 9$^{1/4}$
step 3: (3$^{2/4}$) = 9$^{1/4}$
step 4: (3$^{1/2}$) = 9$^{1/4}$
… final answer: log$_3$ ($^4\surd$9) = 1/2
What’s got me confused is that I arrived at the process in the second half by a flash of insight/intuition and am struggling to determine how the operations I perform on the LHS are legal, since in step 1 I appear to change the value of one side without changing the value of the other side.
I can only conclude that my understanding of equation transposition must be flawed. Request for comment.