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.
