Sure, let's rewrite the given expression [tex]\(\frac{\log \sqrt{x^{17}}}{\log \frac{y}{z^4}}\)[/tex] in terms of the variables [tex]\(u\)[/tex], [tex]\(v\)[/tex], and [tex]\(w\)[/tex], where [tex]\(u = \log x\)[/tex], [tex]\(v = \log y\)[/tex], and [tex]\(w = \log z\)[/tex].
### Step-by-step Solution
1. Simplify the numerator:
The expression in the numerator is [tex]\(\log \sqrt{x^{17}}\)[/tex].
[tex]\[
\log \sqrt{x^{17}} = \log \left(x^{17/2}\right)
\][/tex]
Using the property of logarithms [tex]\(\log (a^b) = b \log a\)[/tex]:
[tex]\[
\log \left(x^{17/2}\right) = \frac{17}{2} \log x
\][/tex]
Since [tex]\(u = \log x\)[/tex], we have:
[tex]\[
\log \sqrt{x^{17}} = \frac{17}{2} u
\][/tex]
2. Simplify the denominator:
The expression in the denominator is [tex]\(\log \frac{y}{z^4}\)[/tex].
Using the property of logarithms [tex]\(\log \frac{a}{b} = \log a - \log b\)[/tex]:
[tex]\[
\log \frac{y}{z^4} = \log y - \log z^4
\][/tex]
Then, using the property [tex]\(\log (a^b) = b \log a\)[/tex]:
[tex]\[
\log z^4 = 4 \log z
\][/tex]
Since [tex]\(v = \log y\)[/tex] and [tex]\(w = \log z\)[/tex], we get:
[tex]\[
\log \frac{y}{z^4} = \log y - 4 \log z = v - 4w
\][/tex]
3. Form the final expression:
Now, we combine the simplified numerator and denominator:
[tex]\[
\frac{\log \sqrt{x^{17}}}{\log \frac{y}{z^4}} = \frac{\frac{17}{2} u}{v - 4w}
\][/tex]
Thus, the expression [tex]\(\frac{\log \sqrt{x^{17}}}{\log \frac{y}{z^4}}\)[/tex] rewritten in terms of [tex]\(u\)[/tex], [tex]\(v\)[/tex], and [tex]\(w\)[/tex] is:
[tex]\[
\frac{\log \sqrt{x^{17}}}{\log \frac{y}{z^4}} = \frac{8.5u}{v - 4w}
\][/tex]
