express in terms log(x) log(y)

**decoy808** · March 6th 2010, 12:21 PM

the question is...

$\frac{1}{2}log(\frac{\sqrt{x}}{xy})$

my attempt...

$\frac{1}{2}log\sqrt{x}-\frac{1}{2}log(xy)$

$log{x}-\frac{1}{2}log(xy)$

the bit what i was unsire about was the sqrt

many thanks

**Quacky** · March 6th 2010, 01:04 PM

Note that $\sqrt x = x^{\frac{1}{2}}$

Also note that $log(xy) = log(x)+log(y)$

So to start, I would say:

$=\frac{1}{2}log(x^\frac{1}{2})-\frac{1}{2}Log(xy)$

How would you continue to simplify?

**decoy808** · March 6th 2010, 03:40 PM

im geting a bit confused with the simplifying.

do i remove the 1/2 by multipying by 2?

and im unsure how i would bring the x of the xy to the other side?

**Quacky** · March 6th 2010, 04:30 PM

Let's split this up and simplify the

$\frac{1}{2}log(x^\frac{1}{2})$ first.

Do you recall that

$Log(a^b)=b\times log(a)$ ?

How would you apply that rule here?

Now let's look at:

$-\frac{1}{2}log(xy)$

$= -\frac{1}{2} [log(xy)]$

Now, to simplify this, why not just apply the

rule?

**decoy808** · March 6th 2010, 04:56 PM

$\frac{1}{2}(-\frac{1}{2}log(x)+log(y))$ ?

**Quacky** · March 6th 2010, 05:03 PM

Edit, nevermind, will edit in a sec.

**Quacky** · March 7th 2010, 02:23 AM

$<br /> =\frac{1}{2}\times\frac{1}{2}Log(x)-\frac{1}{2}[Log(x)+Log(y)]<br />$

$=\frac{1}{4}Log(x)-\frac{1}{2}log(x)-\frac{1}{2}Log(y)$

$=\frac{1}{2}[\frac{1}{2}Log(x)-Log(x)-Log(y)]$

$=\frac{1}{2}[-\frac{1}{2}Log(x)-Log(y)]$

I think that this is correct, I'll check in a second. So you just made a sign error somewhere.

Edit: I've checked, and it seems that I am correct. I must apologize for the late reply - I was caught on my PC when I should have been asleep, so I was forced to turn it off abrutly.

**decoy808** · March 7th 2010, 03:40 AM

thank you for your help

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