I deleted the solution I had posted here because I mistook your given
instructions.  See my solution at the bottom which is a continuation
of what you did.

Edwin

It may be that your instructions were to write this as a single log.
Always be sure to list the exact instructions given:

Usually the instruction for this sort of log problem will be to expand as
far as possible, or to write as a single log. "To simplify" is a bit
ambiguous in log problems of this type.

log(a)-1/2log(4a)
I get this far:
log(a) - log(4a)^1/2
log[a/(4a)^1/2]

Then you can continue this way:







Subtract exponents of a:





I notice now that this is identical with Theo's solution, which
wasn't there before.

Edwin

Using the standard rules of logarithms, solve it this way


      log(a) - (1/2)*log(4a) = log(a) - (1/2) * [log(4) + log(a)] = log(a) - (1/2)*log(4) - (1/2)*log(a)  (*)


Combine the 1st and the 3rd terms in (*)  log(a) - (1/2)*log(a) = (1/2)*log(a).


Transform the 2nd term in (*)              (1/2)*log(4) = (1/2) * (2*log(2)) = log(2).



So, you can continue line (*) in this way

       log(a) - (1/2)*log(4a) = (1/2)*log(a) - log(2).   (1)



The rest depends for whom you do this assignment.


If it is a Math assignment, then you transform right side of (1) into  

        log(a) - (1/2)*log(4a) = .

It gives you the answer in simplest mathematical form.



If you simplify for  "Computer Science",  they prefer another form as a "simplest"

    log(a) - (1/2)*log(4a) = (1/2)*log(a) - log(2),    (2)

same as (1).


Why ? - - - because, firstly, then we do not need calculate ,

                and, secondly, log(2) can be considered as a constant.


So, if there is the need to compute log(a) - (1/2)*log(4a)  many times for different values of 'a'

(as it often happens in computing),  then form (2) is preferable. 


For computing, they do not use the term "simplest form" of an expression.


Instead, they think about the form, which is most    for calculations,
i.e. requires minimum of arithmetic operations.

I have to simplify the following - all base 10 logarithms:
log(a)-1/2log(4a)
I get this far:
log(a) - log(4a)^1/2
log[a/(4a)^1/2]
but then I get stuck!
log(a) - ___ <=== I got this far, but then I get stuck!
log(a) -  <==== Picking up from HERE!!
log(a) - 
log(a) - ()
log(a) - ()
log(a) - 
log 
log () ----- RATIONALIZING denominator
log () ====> log () ====>