of 21 Step 1 of 1 $$ \ln \left(\frac{x^{8} \sqrt{p^{7} q^{3}}}{e^{9}}\right) $$ or. $$ \text { ogarithms to expand the following expression as much as possible. Simplify any numerical expressions that can be evaluat } $$

Question

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1. Write the given expression:
   
   $$
   \ln \left(\frac{x^{8} \sqrt{p^{7} q^{3}}}{e^{9}}\right)
   $$

2. Use the logarithm rule for division, $$\ln \frac{A}{B} = \ln A - \ln B$$:

$$
   \ln \left(\frac{x^{8} \sqrt{p^{7} q^{3}}}{e^{9}}\right) = \ln \left(x^{8} \sqrt{p^{7} q^{3}}\right) - \ln \left(e^{9}\right)
   $$

3. Use the logarithm rule for multiplication, $$\ln (AB) = \ln A + \ln B$$, on the numerator:

$$
   \ln \left(x^{8} \sqrt{p^{7} q^{3}}\right) = \ln \left(x^{8}\right) + \ln \left(\sqrt{p^{7} q^{3}}\right)
   $$

4. Rewrite the square root as an exponent, noting that $$\sqrt{a} = a^{\frac{1}{2}}$$:

$$
   \sqrt{p^{7} q^{3}} = \left(p^{7} q^{3}\right)^{\frac{1}{2}} = p^{\frac{7}{2}}q^{\frac{3}{2}}
   $$
   
   So,
   
   $$
   \ln \left(\sqrt{p^{7} q^{3}}\right) = \ln \left(p^{\frac{7}{2}} q^{\frac{3}{2}}\right)
   $$

5. Expand the logarithm of the product using the rule $$\ln (AB) = \ln A + \ln B$$:

$$
   \ln \left(p^{\frac{7}{2}} q^{\frac{3}{2}}\right) = \ln \left(p^{\frac{7}{2}}\right) + \ln \left(q^{\frac{3}{2}}\right)
   $$

6. Use the logarithm power rule, $$\ln \left(a^b\right) = b \ln a$$:

$$
   \ln \left(x^{8}\right) = 8 \ln(x)
   $$
   $$
   \ln \left(p^{\frac{7}{2}}\right) = \frac{7}{2} \ln(p)
   $$
   $$
   \ln \left(q^{\frac{3}{2}}\right) = \frac{3}{2} \ln(q)
   $$

7. Recognize that $$\ln \left(e^{9}\right) = 9 \ln(e) = 9$$ (since $$\ln(e)=1$$).

8. Combine all the pieces:

$$
   \ln \left(\frac{x^{8} \sqrt{p^{7} q^{3}}}{e^{9}}\right) = 8 \ln(x) + \frac{7}{2} \ln(p) + \frac{3}{2} \ln(q) - 9
   $$

Thus, the expanded expression is:

$$
8 \ln(x) + \frac{7}{2} \ln(p) + \frac{3}{2} \ln(q) - 9
$$
The expanded expression is: $$ 8 \ln(x) + \frac{7}{2} \ln(p) + \frac{3}{2} \ln(q) - 9 $$

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