The rules are ...
[tex]\Longrightarrow \,\, \ln(a\,b)=\ln(a)+\ln(b)[/tex]
[tex]\Longrightarrow \,\, \ln(a^{b})=b\ln(a)[/tex]
Your expression makes use of both of these rules.
[tex]\ln(2m^{8})=\ln(2)+\ln(m^{8})=\ln(2)+8\ln(m)[/tex]
The expanded form is
[tex]\ln(2)+8\ln(m)[/tex]
