## Tuesday, December 15, 2009

### 7.4 Laws of Logarithms

After we've learned the basic properties of logarithms, here are some laws that all log functions abide by!

Product Law of Logs

loga(m*n) = logam + logan

if a>0, so m>0,
for example:
let m = ax
logam = x

and if n>0, then a≠1
let n = ay
logan = y

that would mean...
m*n = ax * ay
m*n = a
x + y
loga(m*n) = x+y

therefore:
the new proven product law!
loga(m*n) = logam + logan

For Example:
log64+ log654

if the law applied then it would mean...
log64+ log654 = log6(4*54)
log64+ log654 = log6(216)
6x = 216
6x = 63
x = 3

therefore:
log64+ log654 = log6(216) = 3

Quotient Law of Logs:

loga(m/n) = logam - logan

so if a>0, then m>0
let m = a
x
logam = x

if n>0, a≠1
let n = ay
logan = y

now...
m / n = ax / ay
m / n = a
x-y
loga (m / n) = x - y

therefore our new proven quotient law is:

loga(m / n) = logam - logan

For example:
log5
375-log515

if the law applied, that would mean:
log5375-log515 = log5(375/15)
log5
375-log515 =
log5(25)
5x = 25
5x =
52
x = 2

that would mean
log5375-log515 = log5(25) = 2