To rewrite the given logarithmic expression $\log\left(\frac{x^{19}y^{16}}{z^{15}}\right)$ as a sum or difference of logarithms with no exponents, follow these steps:

1. Apply the logarithm quotient rule: $\log\left(\frac{A}{B}\right) = \log(A) - \log(B)$ This gives: $\log\left(\frac{x^{19}y^{16}}{z^{15}}\right) = \log(x^{19}y^{16}) - \log(z^{15})$

2. Apply the logarithm product rule: $\log(AB) = \log(A) + \log(B)$ For $\log(x^{19}y^{16})$, this becomes: $\log(x^{19}y^{16}) = \log(x^{19}) + \log(y^{16})$

3. Apply the logarithm power rule: $\log(A^n) = n\log(A)$ Apply this to each term: $\log(x^{19}) = 19\log(x), \quad \log(y^{16}) = 16\log(y), \quad \log(z^{15}) = 15\log(z)$

4. Substitute everything back: $\log\left(\frac{x^{19}y^{16}}{z^{15}}\right) = 19\log(x) + 16\log(y) - 15\log(z)$

Final Answer: $\log\left(\frac{x^{19}y^{16}}{z^{15}}\right) = 19\log(x) + 16\log(y) - 15\log(z)$

Would you like further explanations or any additional examples?

Related Questions:

1. What are the logarithmic rules and how do they apply to simplify complex expressions?
2. How does the logarithm quotient rule work in simplifying equations?
3. What are real-world applications of logarithmic expressions like this one?
4. How would the expression change if the base of the logarithm were specified?
5. Can you explain the logarithmic product and power rules with more examples?

Tip:

Always expand expressions step-by-step to ensure clarity and avoid missing details in