Given the following properties for logarithms:
[tex]\begin{gathered} b\log_{10}a=\log_{10}a^b...(1) \\ \\ \log_{10}a+\log_{10}b=\log_{10}a\cdot b...(2) \\ \\ \operatorname{\log}_{10}a-\operatorname{\log}_{10}b=\operatorname{\log}_{10}\frac{a}{b}...(3) \end{gathered}[/tex]
Then, from the problem, using (1):
[tex]\begin{gathered} 5\log_{10}x+\log_{10}20-\log_{10}10 \\ \\ \log_{10}x^5+\log_{10}20-\log_{10}10 \end{gathered}[/tex]
Now, using (2):
[tex]\log_{10}20\cdot x^5-\log_{10}10=\log_{10}(20x^5)-1[/tex]
Finally, using (3):
[tex]\begin{gathered} \log_{10}\frac{20x^5}{10} \\ \\ \therefore\log_{10}(2x^5) \end{gathered}[/tex]
Answer: Third and last options
