Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Our Q&A platform provides quick and trustworthy answers to your questions from experienced professionals in different areas of expertise. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To solve the expression [tex]\(\log \left(\frac{x^9 y^{16}}{z^{13}}\right)\)[/tex] using properties of logarithms, let's proceed step-by-step:
1. Review the Properties of Logarithms:
- The logarithm of a product: [tex]\(\log(ab) = \log(a) + \log(b)\)[/tex]
- The logarithm of a quotient: [tex]\(\log \left(\frac{a}{b}\right) = \log(a) - \log(b)\)[/tex]
- The logarithm of a power: [tex]\(\log(a^n) = n \log(a)\)[/tex]
2. Apply the Logarithm of a Quotient:
We start with the logarithm of the given fraction:
[tex]\[ \log \left(\frac{x^9 y^{16}}{z^{13}}\right) = \log(x^9 y^{16}) - \log(z^{13}) \][/tex]
3. Apply the Logarithm of a Product:
Next, we apply the property of the logarithm of a product to the numerator:
[tex]\[ \log(x^9 y^{16}) = \log(x^9) + \log(y^{16}) \][/tex]
4. Apply the Logarithm of a Power:
Now we apply the property of the logarithm of a power to each term:
[tex]\[ \log(x^9) = 9 \log(x) \][/tex]
[tex]\[ \log(y^{16}) = 16 \log(y) \][/tex]
[tex]\[ \log(z^{13}) = 13 \log(z) \][/tex]
5. Combine All Steps:
Combine the logs we have applied to get the simplified result:
[tex]\[ \log \left(\frac{x^9 y^{16}}{z^{13}}\right) = \log(x^9 y^{16}) - \log(z^{13}) \][/tex]
[tex]\[ = \left(9 \log(x) + 16 \log(y)\right) - 13 \log(z) \][/tex]
6. Final Answer:
Therefore, the expression simplified using logarithmic properties is:
[tex]\[ \log \left(\frac{x^9 y^{16}}{z^{13}}\right) = 9 \log(x) + 16 \log(y) - 13 \log(z) \][/tex]
This is the detailed, step-by-step solution to the given question.
1. Review the Properties of Logarithms:
- The logarithm of a product: [tex]\(\log(ab) = \log(a) + \log(b)\)[/tex]
- The logarithm of a quotient: [tex]\(\log \left(\frac{a}{b}\right) = \log(a) - \log(b)\)[/tex]
- The logarithm of a power: [tex]\(\log(a^n) = n \log(a)\)[/tex]
2. Apply the Logarithm of a Quotient:
We start with the logarithm of the given fraction:
[tex]\[ \log \left(\frac{x^9 y^{16}}{z^{13}}\right) = \log(x^9 y^{16}) - \log(z^{13}) \][/tex]
3. Apply the Logarithm of a Product:
Next, we apply the property of the logarithm of a product to the numerator:
[tex]\[ \log(x^9 y^{16}) = \log(x^9) + \log(y^{16}) \][/tex]
4. Apply the Logarithm of a Power:
Now we apply the property of the logarithm of a power to each term:
[tex]\[ \log(x^9) = 9 \log(x) \][/tex]
[tex]\[ \log(y^{16}) = 16 \log(y) \][/tex]
[tex]\[ \log(z^{13}) = 13 \log(z) \][/tex]
5. Combine All Steps:
Combine the logs we have applied to get the simplified result:
[tex]\[ \log \left(\frac{x^9 y^{16}}{z^{13}}\right) = \log(x^9 y^{16}) - \log(z^{13}) \][/tex]
[tex]\[ = \left(9 \log(x) + 16 \log(y)\right) - 13 \log(z) \][/tex]
6. Final Answer:
Therefore, the expression simplified using logarithmic properties is:
[tex]\[ \log \left(\frac{x^9 y^{16}}{z^{13}}\right) = 9 \log(x) + 16 \log(y) - 13 \log(z) \][/tex]
This is the detailed, step-by-step solution to the given question.
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.