To simplify $\log\left(\frac{z^{18}y^{13}}{z^{8}}\right)$ as a sum or difference of logarithms with no exponents, follow these steps:

1. Rewrite the expression by separating the numerator and the denominator: $\log\left(\frac{z^{18}y^{13}}{z^{8}}\right) = \log(z^{18} y^{13}) - \log(z^8)$

2. Apply the product property of logarithms on $\log(z^{18} y^{13})$: $= \left(\log(z^{18}) + \log(y^{13})\right) - \log(z^8)$

3. Apply the power rule for logarithms, $\log(a^b) = b \cdot \log(a)$, to each term: $= 18 \cdot \log(z) + 13 \cdot \log(y) - 8 \cdot \log(z)$

4. Combine like terms by grouping the $\log(z)$ terms: $= (18 - 8) \cdot \log(z) + 13 \cdot \log(y)$ $= 10 \cdot \log(z) + 13 \cdot \log(y)$

So, the simplified form is: $10 \cdot \log(z) + 13 \cdot \log(y)$

Would you like more details on any step?

Here are 5 related questions for further exploration:

1. How does the power rule for logarithms work?
2. How can the quotient property of logarithms simplify similar expressions?
3. What is the product rule in logarithmic expressions?
4. How would the expression change if the denominator was $y^8$ instead of $z^8$?
5. How are logarithmic properties used to solve equations with exponential expressions?

Tip: Remember, the power rule for logarithms is very useful for breaking down exponents within logarithmic expressions!