Proofs of Logarithm Properties
logaM+logaN=logaMN (1) proof: assume: logaM=m,logaN=n so: am=M,an=N⇒M⋅N=am⋅an=am+n logaMN=m+n=logaM+logaN logaM−logaN=logaMN (2) proof: assume: logaM=m,logaN=n so: am=M,an=N⇒MN=aman=am−n⇒logaMN=m−n=logaM−logaN logaaM=M (3) proof: assume: aM=B⇒logaB=M so: logaaM=M alogaM=M (4) proof: assume: logaM=B so: aB=M ∵B=logaM,aB=M ∴aB=alogaM=M logaMN=NlogaM (5) proof: logaMN=loga(N⏞M⋅M⋯M) ∵logaM+logaN=logaMN property (1) ∴loga(N⏞M⋅M⋯M)=N⏞logaM+logaM+⋯logaM=NlogaM logab=logcblogca=lnblna=lgblga (6) ...