Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Get immediate answers to your questions from a wide network of experienced professionals on our Q&A platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
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 your visit. We're dedicated to helping you find the information you need, whenever you need it. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
