Rewriting a log

1. Jun 29, 2008

3.141592654

1. The problem statement, all variables and given/known data

Use the Laws of logarithms to rewrite the expression in a form with no logarithm of a product, quotient or power.
ln (x^10*sqrt((y^19)/(z^7))) = a ln(x)+b ln(y)+c ln(z)

2. Relevant equations

ln (x^a)=a ln(x)

3. The attempt at a solution

I know that it will start =10 ln(x), but I don't know what the square root implies. To specify, if I had the equation ln(sqrt(xy)), I thought the answer would be ln(x)+2 ln(y), but this isn't the case. Can anyone explain what happens to the square root when rewriting this expression? Thanks for your help!

2. Jun 29, 2008

chroot

Staff Emeritus
Taking the square root of a quantity is the same as raising that quantity to the 1/2th power. In other words, the square root is interchangeably just an exponent of 1/2.

- Warren

3. Jun 30, 2008

HallsofIvy

Staff Emeritus
Another very relevant equation for this problem is ln(ab)= ln(a)+ ln(b).

As chroot told you sqrt is "1/2" power. Notice that even if you had 2nd power, you would NOT have ln((xy)2)= ln(x)+ 2 ln(y). ln((xy)[2= 2ln(xy)= 2[ln(x)+ ln(y)]= 2ln(x)+ 2ln(y).

Now, ln(sqrt(xy))= ln((xy)1/2)= what?

4. Jun 30, 2008

3.141592654

thanks for your help, both of you, I was able to figure out the problem with this!
Halls of Ivy,
ln(sqrt(xy))= ln((xy)1/2)= 1/2ln(xy)= 1/2ln(x)+ 1/2ln(y)

5. Jun 30, 2008

HallsofIvy

Staff Emeritus
Exactly!