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Important properties of logarithms:logb1= logbb= logbbx= blogbx= Since logarithms are exponents, the laws of exponents give corresponding laws of logarithms.Law of LogarithmsEquivalent Exponent RuleProduct Law of Logarithmslogb(MN)=logbM+logbNQuotient Law of LogarithmslogbMN=logbM-logbNPower Law of Logarithmslogbxn=nlogbxExamples of how to use each law of logarithm:Example 1Evaluate.log42+log432b) log2144-log29 c) log0.0012Proof of the Laws of LogarithmsThe Product Law of Logarithms proof:(Proof of logb(MN)=logbM+logbN) Let logbM=m and logbN=nThe Quotient Law of Logarithms proof:(Proof of logbMN=logbM-logbN)The Power Law of Logarithms proof:(Proof of logbxn=nlogbx)The power law is also used to evaluate logs with bases other than 10.Example 2Evaluate 2=1.05xChange of Base FormulaMany calculators cannot calculate logs of bases other than 10 or e. We can use the change of base formula to calculate logarithms of different bases.lognm=logbmlogbn b>0, b≠1, m>0, n>0Proof of the change of base formula:Example 3Evaluate log241 correctly to 3 decimal places.Example 4 Evaluate each and state the rule(s) used. b) c) d) f) g) h) i) j) Example 5 Write as a single log b) c) d) log63+12log65-log62 e) logab+loga7c+loga4b-logac ................
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