To simplify the given logarithmic expression $\log \left( \frac{x^{18}y^2}{z^{17}} \right)$ as a sum or difference of logarithms without exponents:

Step-by-Step Solution:

1. Apply the logarithmic quotient rule: $\log \left( \frac{A}{B} \right) = \log(A) - \log(B)$ This gives: $\log \left( \frac{x^{18}y^2}{z^{17}} \right) = \log(x^{18}y^2) - \log(z^{17}).$

2. Apply the logarithmic product rule to $\log(x^{18}y^2)$: $\log(AB) = \log(A) + \log(B)$ This gives: $\log(x^{18}y^2) = \log(x^{18}) + \log(y^2).$

3. Simplify each term using the power rule: $\log(A^n) = n\log(A)$

   • For $\log(x^{18})$: $\log(x^{18}) = 18\log(x)$,
   • For $\log(y^2)$: $\log(y^2) = 2\log(y)$,
   • For $\log(z^{17})$: $\log(z^{17}) = 17\log(z)$.

4. Combine all terms: Substitute these back into the expression: $\log \left( \frac{x^{18}y^2}{z^{17}} \right) = \left[ 18\log(x) + 2\log(y) \right] - 17\log(z).$

   Simplify: $\log \left( \frac{x^{18}y^2}{z^{17}} \right) = 18\log(x) + 2\log(y) - 17\log(z).$

Final Answer:

$\log \left( \frac{x^{18}y^2}{z^{17}} \right) = 18\log(x) + 2\log(y) - 17\log(z).$

Would you like further clarification or additional examples?

Related Questions:

1. How do you simplify logarithmic expressions with more variables?
2. Can you explain the difference between the product rule and the quotient rule for logarithms?
3. What happens when logarithmic bases are different? How do you simplify?
4. How do you deal with logarithmic expressions involving roots (e.g., $\log(\sqrt{x})$)?
5. What are some practical applications of logarithmic simplifications?

Tip:

Always apply the power rule last to eliminate exponents in logarithmic expressions!