IDNLearn.com provides a collaborative environment for finding and sharing knowledge. Join our knowledgeable community and access a wealth of reliable answers to your most pressing questions.
Sagot :
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]
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]
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Thank you for visiting IDNLearn.com. We’re here to provide dependable answers, so visit us again soon.
