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Proofs of Logarithm Properties

logaM+logaN=logaMN  (1) proof:  assume: logaM=m,logaN=n  so:  am=M,an=NMN=aman=am+n  logaMN=m+n=logaM+logaN  logaMlogaN=logaMN (2) proof: assume: logaM=m,logaN=n so: am=M,an=NMN=aman=amnlogaMN=mn=logaMlogaN logaaM=M (3) proof: assume: aM=BlogaB=M so: logaaM=M alogaM=M (4) proof: assume: logaM=B so: aB=M \therefore a^B = a^{log_aM} = M log_aM^N = Nlog_aM (5) proof: log_aM^N = log_a(\overbrace{M \cdot M \cdots M}^{N}) \because log_aM +  log_aN = log_aMN  property (1) \therefore log_a(\overbrace{M \cdot M \cdots M}^{N}) = \overbrace{log_aM + log_aM + \cdots log_aM}^{N} = Nlog_aM log_ab = \frac{log_cb}{log_ca}= \frac{lnb}{lna} = \frac{lgb}{lga} (6) ...