Use the properties of logarithms to expand the expression. \( \ln \left(5 m^{6}\right) \) \( \ln \left(5 m^{6}\right)=\square \)

(6) \( \frac{9}{2} \sqrt{\frac{5}{3}}= \)

Use the properties of logarithms to expand the following expression. \[ \log \left(\frac{x^{5}}{\sqrt{(x+6)^{3}}}\right) \] Your answer should not have radicals or exponents. You may assume that all variables are positive.

3. \( \frac{1}{2} \cdot \log _{3} 81 \)

Complete the conjecture that describes the pattern in the sequence. \( 4,8,12,16,20 \)

Solve the given exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation for the solution. \( 3 e^{\mathrm{x}}=53 \) The solution set expressed in terms of logarithms is (Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any numbers in the equation.) Now use a calculator to obtain a decimal approximation for the solution. The solution set is \( \{\square\} \). (Use a comma to separate answers as needed. Round to two decimal places as needed.)