Basic properties that will be used:
[tex]\log_ab^n=n\log_ab[/tex]
[tex]x^{1/n}=\sqrt[n]x[/tex]
[tex]\log_aa=1[/tex]
All of these expression can be written to match on or more of the patterns above.
[tex]19.\quad\log_5\sqrt[4]{25}=\log_5(5^2)^{1/4}=\dfrac14\log_55^2=\dfrac12\log_55=\dfrac12[/tex]
[tex]21.\quad9\ln e^3+4\ln e^5=27\ln e+20\ln e=27+20=47[/tex]
[tex]23.\quad2\log_3\sqrt{27}=2\log_3(3^3)^{1/2}=\log_33^3=3\log_33=3[/tex]
[tex]25.\quad4\log_2\sqrt8=4\log_2(2^3)^{1/2}=2\log_22^3=6\log_22=6[/tex]
[tex]27.\quad\log_3\sqrt[6]{243}=\log_3(3^5)^{1/6}=\dfrac16\log_33^5=\dfrac56\log_33=\dfrac56[/tex]
