Explaining the non-application of the multiplication law of logarithms, when logs are in the denominators. I have an A' Levels student who had to solve the following problem:
$ log_2 x + log_4 x = 2$
This was to be solved using the Change of base rule, and then substitution, as follows:
$ \frac{1}{log_x 2} + \frac{1}{log_x 4} = 2$
=>
$ \frac{1}{log_x 2} + \frac{1}{log_x 2^2} = 2$
=>
$ \frac{1}{log_x 2} + \frac{1}{2 log_x 2} = 2$
=>
$ \frac{1}{y} + \frac{1}{2y} = 2$
and so on.
My student has this confusion:
Given that $log_x 2 + log_x 4 = log_x 8$ why can't we go from 
$ \frac{1}{log_x 2} + \frac{1}{log_x 4} = 2$
directly to:
$ \frac{1}{log_x 8} = 2$
I'm having a hard time explaining why... I thought of trying to explain the issue using laws of fraction arithmetic, is that right? What is the answer? 

You could try an example. Is it true that
$$\frac12+\frac12=\frac1{2+2}=\frac14\mathrm{?}$$
