SOLUTION: express as a single log: 1/2loga x + 4loga y -1/3loga z

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Question 107094: express as a single log: 1/2loga x + 4loga y -1/3loga z
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
Let's look at what we know about logarithms:
loga%28x%5En%29=n%2Aloga%28x%29,

loga%28x%29+%2B+loga%28y%29+=+loga%28x%2Ay%29

loga%28x%29+-+loga%28y%29+=+loga%28x%2Fy%29
Using these facts, let's operate on your orignial expression:
%281%2F2%29loga%28x%29+%2B+4%2Aloga%28y%29+-%281%2F3%29loga%28z%29

Beginning with the first term, remember that x%5E%281%2F2%29=sqrt%28x%29, so

%281%2F2%29loga%28x%29=loga%28sqrt%28x%29%29

Similarly, the other two terms become:

4%2Aloga%28y%29=loga%28y%5E4%29, and
%281%2F3%29loga%28z%29+=+loga%28z%5E%281%2F3%29%29
(z%5E%281%2F3%29 is the cube root of z, but I can't render that on this system)

Putting the terms together, using the second rule from above:

loga%28%28y%5E4%29%2Asqrt%28x%29%2Fz%5E%281%2F3%29%29

The answer would be much neater if I could replace the fractional power on z with a cube root radical, but it is still correct in this form.