6.4 Power of Law of Logarithms

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.