[tex]\mathsf{log_{23}[log_5(log_232)]}\\\\\\\mathsf{Step~one,~find~log_232:}\\\\\\\mathsf{log_232=y}\\\\\\\mathsf{2^y=32~~~~(factor~the~32)}\\\\\\\mathsf{2^y=2^5}\\\\\\\mathsf{y=5}\\\\\\\mathsf{log_232~replaced~by~5}\\\\\\\mathsf{log_{23}(log_55)}[/tex]
[tex]\mathsf{Step~two,~find~log_55:}\\\\\\\mathsf{log_55=x}\\\\\\\mathsf{5^x=5}\\\\\\\mathsf{x=1}[/tex]
[tex]\mathsf{log_55~replaced~by~1:}\\\\\\\mathsf{log_{23}1}\\\\\\\mathsf{log_{23}1=z}\\\\\\\mathsf{23^z=1}\\\\\\\mathsf{z=0}\\\\\\\\\mathsf{Therefore:~~log_{23}[log_5(log_232)]=z=0}\\\\\\\large\fbox{$\mathsf{log_{23}[log_5(log_232)]=0}$}[/tex]
