To expand the expression [tex]\(\ln \left(\frac{x^3 z^5}{y^7}\right)\)[/tex] using logarithm properties, we can follow these steps:
1. Apply the Logarithm of a Quotient Property:
The property states that [tex]\(\ln \left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)[/tex].
Applying this property, we get:
[tex]\[
\ln \left(\frac{x^3 z^5}{y^7}\right) = \ln(x^3 z^5) - \ln(y^7)
\][/tex]
2. Apply the Logarithm of a Product Property:
The property states that [tex]\(\ln(ab) = \ln(a) + \ln(b)\)[/tex].
Applying this property to [tex]\(\ln(x^3 z^5)\)[/tex], we get:
[tex]\[
\ln(x^3 z^5) = \ln(x^3) + \ln(z^5)
\][/tex]
3. Apply the Logarithm of a Power Property:
The property states that [tex]\(\ln(x^k) = k \cdot \ln(x)\)[/tex].
Applying this property, we get:
[tex]\[
\ln(x^3) = 3 \cdot \ln(x)
\][/tex]
[tex]\[
\ln(z^5) = 5 \cdot \ln(z)
\][/tex]
[tex]\[
\ln(y^7) = 7 \cdot \ln(y)
\][/tex]
4. Combine All the Expanded Terms:
Substitute the expanded logarithms back into the expression:
[tex]\[
\ln(x^3 z^5) - \ln(y^7) = (3 \cdot \ln(x) + 5 \cdot \ln(z)) - 7 \cdot \ln(y)
\][/tex]
5. Simplify the Expression:
Combine the terms to obtain the final expanded expression:
[tex]\[
\ln \left(\frac{x^3 z^5}{y^7}\right) = 3 \cdot \ln(x) + 5 \cdot \ln(z) - 7 \cdot \ln(y)
\][/tex]
Therefore, the expanded expression is:
[tex]\[
3 \cdot \ln(x) + 5 \cdot \ln(z) - 7 \cdot \ln(y)
\][/tex]
