Use main logarithm properties:
[tex]\log_ab^k=k\log_ab;\\ \\\log_ab+\log_ac=\log_abc;\\ \\\log_ab-\log_ac=\log_a\dfrac{b}{c}.[/tex]
Then
1.
[tex]\dfrac{1}{4}\ln x=\ln x^{\frac{1}{4}}.[/tex]
2.
[tex]\dfrac{3}{10}\ln (x+2)=\ln (x+2)^{\frac{3}{10}}.[/tex]
3.
[tex]\ln (x-2)-\dfrac{3}{10}\ln (x+2)=\ln \dfrac{x-2}{(x+2)^{\frac{3}{10}}}.[/tex]
4.
[tex]5\left(\ln (x-2)-\dfrac{3}{10}\ln (x+2)\right)=5\ln \dfrac{x-2}{(x+2)^{\frac{3}{10}}}=\ln \dfrac{(x-2)^2}{(x+2)^{\frac{3}{2}}}.[/tex]
5.
[tex]\dfrac{1}{4}\ln x+5\left(\ln (x-2)-\dfrac{3}{10}\ln (x+2)\right)=\ln x^{\frac{1}{4}}+\ln \dfrac{(x-2)^2}{(x+2)^{\frac{3}{2}}}=\ln \dfrac{x^{\frac{1}{4}}(x-2)^5}{(x+2)^{\frac{3}{2}}}=\\ \\=\ln \dfrac{\sqrt[4]{x}(x-2)^5}{\sqrt{(x+2)^3}}.[/tex]
Answer: correct choice is B
