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 ∵ \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) ...