To expand the expression [tex]\(\log \left(\frac{x^5}{\sqrt[3]{y^2 z}}\right)\)[/tex] using properties of logarithms, we will follow these steps:
1. Use the quotient rule of logarithms:
The quotient rule states that [tex]\(\log \left(\frac{A}{B}\right) = \log(A) - \log(B)\)[/tex]. Applying this, we get:
[tex]\[
\log \left(\frac{x^5}{\sqrt[3]{y^2 z}}\right) = \log(x^5) - \log(\sqrt[3]{y^2 z})
\][/tex]
2. Use the power rule of logarithms on [tex]\(\log(x^5)\)[/tex]:
The power rule states that [tex]\(\log(A^k) = k \log(A)\)[/tex]. Applying this rule to [tex]\(\log(x^5)\)[/tex], we get:
[tex]\[
\log(x^5) = 5 \log(x)
\][/tex]
3. Simplify the denominator inside the logarithm:
Recall that [tex]\(\sqrt[3]{y^2 z}\)[/tex] is the same as [tex]\((y^2 z)^{1/3}\)[/tex]. Therefore:
[tex]\[
\log(\sqrt[3]{y^2 z}) = \log((y^2 z)^{1/3})
\][/tex]
4. Use the power rule of logarithms again:
Apply the power rule [tex]\(\log(A^k) = k \log(A)\)[/tex] to [tex]\(\log((y^2 z)^{1/3})\)[/tex], we get:
[tex]\[
\log((y^2 z)^{1/3}) = \frac{1}{3} \log(y^2 z)
\][/tex]
5. Use the product rule of logarithms:
The product rule states that [tex]\(\log(AB) = \log(A) + \log(B)\)[/tex]. Applying this to [tex]\(\log(y^2 z)\)[/tex], we get:
[tex]\[
\log(y^2 z) = \log(y^2) + \log(z)
\][/tex]
6. Use the power rule of logarithms on [tex]\(\log(y^2)\)[/tex]:
Applying [tex]\(\log(A^k) = k \log(A)\)[/tex] to [tex]\(\log(y^2)\)[/tex], we get:
[tex]\[
\log(y^2) = 2 \log(y)
\][/tex]
Therefore:
[tex]\[
\log(y^2 z) = 2 \log(y) + \log(z)
\][/tex]
7. Combine the results:
Substitute back to get:
[tex]\[
\frac{1}{3} \log(y^2 z) = \frac{1}{3} [2 \log(y) + \log(z)] = \frac{2}{3} \log(y) + \frac{1}{3} \log(z)
\][/tex]
8. Substitute everything back into the original expression:
Now combine all the results from the steps above:
[tex]\[
\log \left(\frac{x^5}{\sqrt[3]{y^2 z}}\right) = 5 \log(x) - \left(\frac{2}{3} \log(y) + \frac{1}{3} \log(z)\right)
\][/tex]
Therefore, we can distribute the negative sign:
[tex]\[
\log \left(\frac{x^5}{\sqrt[3]{y^2 z}}\right) = 5 \log(x) - \frac{2}{3} \log(y) - \frac{1}{3} \log(z)
\][/tex]
Hence, the expanded form of the given expression is:
[tex]\[
\log \left(\frac{x^5}{\sqrt[3]{y^2 z}}\right) = 5 \log(x) - 0.666666666666667 \log(y) - 0.333333333333333 \log(z)
\][/tex]
