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without using a calculator, find the exact value, show work.


log_23[log_5(log_2 32)]


*Log with _ after means subscript.

Verified Answer :

FuturoEngenheiro

\(\mathsf{log_{23}[log_5(log_232)]}\\\\\\\mathsf{Step~one,~find~log_232:}\\\\\\\mathsf{log_232=y}\\\\\\\mathsf{2^y=32~~~~(factor~the~32)}\\\\\\\mathsf{2^y=2^5}\\\\\\\mathsf{y=5}\\\\\\\mathsf{log_232~replaced~by~5}\\\\\\\mathsf{log_{23}(log_55)}\)


\(\mathsf{Step~two,~find~log_55:}\\\\\\\mathsf{log_55=x}\\\\\\\mathsf{5^x=5}\\\\\\\mathsf{x=1}\)



\(\mathsf{log_55~replaced~by~1:}\\\\\\\mathsf{log_{23}1}\\\\\\\mathsf{log_{23}1=z}\\\\\\\mathsf{23^z=1}\\\\\\\mathsf{z=0}\\\\\\\\\mathsf{Therefore:~~log_{23}[log_5(log_232)]=z=0}\\\\\\\large\fbox{$\mathsf{log_{23}[log_5(log_232)]=0}$}\)

log₂₃[log₅(log₂ 32)]

1) We calculate log₂ 32
remember: 32=2⁵; then:
log₂ 32=log₂ 2⁵=5      (log_a a^n=n)

2) We have now:
log₅ (log₂ 32)=log₅ 5=1     (log_a a^n=n)

3) we have now:
log₂₃ [log_5(log_2 32)]=log₂₃ 1=0     (log_a 1=0 ⇔a⁰=1)

Answer: log_23[log_5(log_2 32)]=0