Rewriting Expression w/Log Laws: x^10*sqrt((y^19)/(z^7))

[3.141592654]

Homework Statement

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)

Homework Equations

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

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!

 

[chroot]

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

 

[HallsofIvy]

Homework Statement

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)

Homework Equations

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

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

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!

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?

 

[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)

 

[HallsofIvy]

Exactly!