### Scribin: Laws of Logarithms

Hello, its Cheeks again or will, I will be the scribe for today since my other fellow classmate was suppose to be scribe but did not show up. So here it is:

We started the morning by doing our Dictionaries:

Laws of Logarithms

Product Law: Log_{a} MN = Log_{a} M + Log_{a} N

Example: Log_{23x= Loga3 + Logax}_{Remember: When multiplying powers with the samebase we add the exponents and a logarithm is an exponent.Quotient Law: Loga (M/N) = LogaM - LogaNExample: Log}_{3(5/6) = Log}_{35 - Log}_{36Remember: When dividing powers with the samebase we subtract the exponents and a logarithm is an exponent.Power Law: Log}_{aMc = CLog}_{aMExample: Log 42 = 2Log 4Remember: When a powers is raised to another exponent, we multiply theexponents and a logarithm is an exponent.Change of Base Law: Log}_{aM = Log}_{cM/Log}_{aMExample: Log}_{35 = Log 5/Log 3Special cases:1. aLog}_{ax = x }_{7Log}_{7x = x2. Logaax = x }_{Log33x = x3. Loga1 = 0 Log51 = 04. Logaa = 0 Loga4 = 0We then took a break and did one question on the board:1/2 - Log}_{16(x-3) = Log16x1/2 = Log16x + Log16(x-3)161/2 = Log16x(x-3)4 = x(x-3)0 = x2 - 3x - 40 = (x-4)(x+1)x = 4, x = -1We accept 4 because when we check it with x our Log is positive and we reject -1 because our Log is negative.Proof of Product Law:LetLogaM=x Therefore ax = MLogaN=x Therefore ax = NProve: Loga MN = LogaM + LogaN= LogaMN = (Logax)(Logay)= Logaax + y= x + y= LogaM + LogaNQ.E.D.Proof of Quotient Law:LetLogaM = x therefore ax = MLogaN = y therefore ay = NProve: Loga(M/N) = LogaM - LogaN= Loga(M/N) = Loga(ax/ay)= Logaa(x-y)= x - y= LogaM - LogaNQ.E.D.Proof of the Power Law:LetLogaM =x therefore ax = MProve: Log aMc = CLogaM= LogaMc = Logaacx= cx= CLogaMQ.E.D.Proof of the Change of Base Law:LetLogaM =x therefore ax = MProve: LogaM = LogbM/Logaaax = MLogbax = LogbMx Logba/Logba = LogbM/LogbaQ.E.D. Compound Interest FormulaA = P(1 + r/n)tnA is the amount at the end of the investment period.P is the principal(original) amount invested.r is the percent rate, written as a decimal.n is the number of compounding periods in one year.t is time in years.Interest Compounded ContinuouslyWhen interest is compounded continuously we use this formulaA = PertA is the amount at the end of the investment period.P is the principal(original) amount invested.e is 2.7118281828459r is the percent rate, written as a decimal.t is time in years.So thats our lesson today boys and girls... our next scribe is the one who didnt show up today.... tenn}

please clear me on power law.

ReplyDelete