To write the logarithm [tex]\(\log_5\left(\frac{x^{11}}{yz}\right)\)[/tex] as a sum or difference of logarithms and simplify each term as much as possible, follow these steps:
1. Start with the given logarithm:
[tex]\[
\log_5\left(\frac{x^{11}}{yz}\right)
\][/tex]
2. Use the property of logarithms for division:
[tex]\[
\log_5\left(\frac{a}{b}\right) = \log_5(a) - \log_5(b)
\][/tex]
Applying this property:
[tex]\[
\log_5\left(\frac{x^{11}}{yz}\right) = \log_5(x^{11}) - \log_5(yz)
\][/tex]
3. Use the property of logarithms for multiplication:
[tex]\[
\log_5(ab) = \log_5(a) + \log_5(b)
\][/tex]
Applying this property to [tex]\(\log_5(yz)\)[/tex]:
[tex]\[
\log_5\left(x^{11}\right) - \log_5(yz) = \log_5(x^{11}) - (\log_5(y) + \log_5(z))
\][/tex]
4. Use the property of logarithms for exponents:
[tex]\[
\log_5(a^n) = n \log_5(a)
\][/tex]
Applying this property to [tex]\(\log_5(x^{11})\)[/tex]:
[tex]\[
\log_5(x^{11}) = 11 \log_5(x)
\][/tex]
5. Combine all the simplified terms:
[tex]\[
\log_5\left(\frac{x^{11}}{yz}\right) = 11 \log_5(x) - \log_5(y) - \log_5(z)
\][/tex]
Therefore, the simplified form of the given logarithm is:
[tex]\[
\boxed{11 \log_5(x) - \log_5(y) - \log_5(z)}
\][/tex]
