Hey

Today we learned about Logarithms and the Power Laws.

The basic properties of logarithms are

e.g. log(b) 1 = - <-------> b^0 = 1

The Laws of Logarithms are as follows.

note* log base "a"(or any variable) = log(a)

1. Log(a) mn = log(a)m + log(a)n

Proof:

let: m=a^x, n=a^y

log(a)m=x, log(a)n=y

so:

mn = a^x multiplied by a^y

mn= a^(x+y)

which means that:

log(a) mn = x+y

therefore,

log(a) mn = log(a)m + log(a)n

2. log(a) (m/n) = log(a)m - log(a)n, note n cannot equal zero

let: m=a^x, n=a^y

log(a)m=x, log(a)n=y

so, m/n = a^x/a^y

m/n = a^(x-y)

log(a) (m/n) = x-y

log (a) (m/n) = log(a)m - log(a)n

3. log(a) (n^m) = mlog(a)n

let n=a^x

log(a)n = x

n^m(a^x)^m

n^m = a^(xm)

log(a) (n^m)= xm

log(a) (n^m) = mlog(a)n

4. Change of Base Formula

log(b)X = log(a)X/log(a)b

e.g. a)

3^x = 5

log(3)5 = x

x = log5/log3

x =(approx) 1.4650

Therefore, these following formula's outlines the logarithm laws.

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