To express [tex]\(\log \left(x^2 y^6 z^9\right)\)[/tex] as a sum of logarithms, we can apply the properties of logarithms step-by-step.
1. Use the property of logarithms for products:
[tex]\[
\log(a \cdot b \cdot c) = \log(a) + \log(b) + \log(c)
\][/tex]
For your expression, [tex]\(a = x^2\)[/tex], [tex]\(b = y^6\)[/tex], and [tex]\(c = z^9\)[/tex], so:
[tex]\[
\log \left(x^2 y^6 z^9\right) = \log(x^2) + \log(y^6) + \log(z^9)
\][/tex]
2. Use the power rule for logarithms:
[tex]\[
\log(a^m) = m \cdot \log(a)
\][/tex]
Apply this rule to each term in the sum:
[tex]\[
\log(x^2) = 2 \cdot \log(x)
\][/tex]
[tex]\[
\log(y^6) = 6 \cdot \log(y)
\][/tex]
[tex]\[
\log(z^9) = 9 \cdot \log(z)
\][/tex]
3. Combine these results:
[tex]\[
\log \left(x^2 y^6 z^9\right) = 2 \cdot \log(x) + 6 \cdot \log(y) + 9 \cdot \log(z)
\][/tex]
Therefore, the final expression is:
[tex]\[
\log \left(x^2 y^6 z^9\right) = 2 \log (x) + 6 \log (y) + 9 \log (z)
\][/tex]
So, the values to fill in the blanks are:
[tex]\[
\log \left(x^2 y^6 z^9\right) = \boxed{2} \log (x) + \boxed{6} \log (y) + \boxed{9} \log (z)
\][/tex]
