Answer
$\log\sqrt{x\sqrt{y\sqrt{z}}}=\dfrac{1}{2}\log x+\dfrac{1}{4}\log y+\dfrac{1}{8}\log z$
Work Step by Step
$\log\sqrt{x\sqrt{y\sqrt{z}}}$
Let's rewrite the expression like this:
$\log(x\sqrt{y\sqrt{z}})^{1/2}$
Let's take the exponent to the front of the logarithm to multiply:
$\log(x\sqrt{y\sqrt{z}})^{1/2}=\dfrac{1}{2}\log(x\sqrt{y\sqrt{z}})=...$
The logarithm of a product can be expanded as a sum:
$...=\dfrac{1}{2}[\log x+\log\sqrt{y\sqrt{z}}]=...$
Let's rewrite the expression:
$...=\dfrac{1}{2}[\log x+\log(y\sqrt{z})^{1/2}]=...$
We can take the exponent in $\log(y\sqrt{z})^{1/2}$ to multiply to the front of the logarithm:
$...=\dfrac{1}{2}[\log x+\dfrac{1}{2}\log(y\sqrt{z})]=...$
Again, we expand the logarithm of the product as a sum:
$...=\dfrac{1}{2}[\log x+\dfrac{1}{2}(\log y+\log\sqrt{z})]=...$
Rewrite the expression like this:
$...=\dfrac{1}{2}[\log x+\dfrac{1}{2}(\log y+\log z^{1/2})]=...$
Finally, take the exponent in $\log z^{1/2}$ to multiply to the front of the logarithm:
$...=\dfrac{1}{2}[\log x+\dfrac{1}{2}(\log y+\dfrac{1}{2}\log z)]=...$
$...=\dfrac{1}{2}\log x+\dfrac{1}{4}\log y+\dfrac{1}{8}\log z$
