_{8}(x

^{2})/(y*z

^{2})

- Product rule: log (a*b) = log (a) + log (b)
- Division rule: log (a/b) = log (a) - log (b) (ORDER MATTERS!)
- Exponent rule: lob (a
^{2}) = log (a)^{2}= 2*log (a) (This is NOT the same as (log a)^{2}!!!)

Now that we have those rules, let's apply them to this question. Start with the math operation that is connecting all parts of (x^{2})/(y*z^{2}) together: the division sign.

Apply Rule #2:

log_{8} (x^{2})/(y*z^{2}) --> log_{8 }(x^{2}) - log_{8} (y*z^{2})

Now we have two separate logs to work with. Let's focus on log_{8 }(x^{2}) first.

Apply Rule #3:

log_{8} (x^{2}) --> 2*log_{8} (x)

Now for the second part: log_{8} (y*z^{2}).

Apply Rule #1:

log_{8} (y*z^{2}) --> log_{8} (y) + log_{8} (z^{2})

If you notice, this part can be simplified further. Apply Rule #3 to the log containing z^{2}.

log_{8} (y*z^{2}) --> log_{8} (y) + log_{8} (z^{2}) --> log_{8} (y) + 2*log_{8} (z)

Now that everything has been simplified down as far as we can go, let's update the whole expression:

log_{8} (x^{2})/(y*z^{2}) --> log_{8} (x^{2}) - log_{8} (y*z^{2}) --> 2*log_{8} (x) - [log_{8} (y) + 2*log_{8} (z)]

If you're comparing this answer to the answers in the back of a textbook, they may take this one step further and distribute the subtraction sign through to the log_{8}(y) and the 2log_{8}(z), in which case, your answer would look like this:

2*log_{8} (x) - log_{8} (y) - 2*log_{8} (z)

Hope this is helpful!!

~Patty