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<sub>23x= Loga3 + Logax
Remember: When multiplying powers with the same
base we add the exponents and a logarithm is an exponent.

Quotient Law: Loga (M/N) = LogaM - LogaN
Example: Log3(5/6) = Log35 - Log36
Remember: When dividing powers with the same
base we subtract the exponents and a logarithm is an exponent.

Power Law: LogaM^c = CLogaM
Example: Log 4² = 2Log 4
Remember: When a powers is raised to another exponent, we multiply the
exponents and a logarithm is an exponent.

Change of Base Law: LogaM = LogcM/LogaM
Example: Log35 = Log 5/Log 3
Special cases:
1. a^Logax = x 7^Log7x = x
2. Loga^ax = x Log3^3x = x
3. Loga1 = 0 Log51 = 0
4. Logaa = 0 Loga4 = 0

We then took a break and did one question on the board:

1/2 - Log16(x-3) = Log16x
1/2 = Log16x + Log16(x-3)
16^1/2 = Log16x(x-3)
4 = x(x-3)
0 = x² - 3x - 4
0 = (x-4)(x+1)
x = 4, x = -1

We 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:
Let
LogaM=x Therefore a^x = M
LogaN=x Therefore a^x = N
Prove: Loga MN = LogaM + LogaN
= LogaMN = (Loga^x)(Loga^y)
= Logaa^{x + y}
= x + y
= LogaM + LogaN
Q.E.D.

Proof of Quotient Law:
Let
LogaM = x therefore ax = M
LogaN = y therefore ay = N
Prove: Loga(M/N) = LogaM - LogaN
= Loga(M/N) = Loga(ax/ay)
= Logaa^(x-y)
= x - y
= LogaM - LogaN
Q.E.D.

Proof of the Power Law:
Let
LogaM =x therefore a^x = M
Prove: Log aM^c = CLogaM
= LogaM^c = Logaa^cx
= cx
= CLogaM
Q.E.D.

Proof of the Change of Base Law:
Let
LogaM =x therefore a^x = M
Prove: LogaM = LogbM/Logaa
a^x = M
Logba^x = LogbM
x Logba/Logba = LogbM/Logba
Q.E.D.

Compound Interest Formula

A = P(1 + r/n)^tn

A 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 Continuously

When interest is compounded continuously we use this formula

A = Pe^rt

A is the amount at the end of the investment period.
P is the principal(original) amount invested.
e is 2.7118281828459
r 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</sub>