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

log_23[log_5(log_2 32)]

*log with _ after means subscript.

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)

log_23[log_5(log_2 32)]=0