Tuesday, December 15, 2009

7.4 Laws of Logarithms

Exponent Laws we have learned in previous years, translate into the Laws of Logarithms.

1. Product Law of Logarithms
loga(m*n) = logam + logan
a>0, m>0, n>0, a≠1

Proof:
let m = ax
let n = a
y

Now...turn these into logarithms:
logam = x
log
an = y

So...
m*n = ax * ay
m*n = a
x+y
loga(m*n) = x+y
loga(m*n) = logam + logan

Example:
Evaluate... log84 + log816
log8(4*16)
log864
8x=64
8x=82
x=2

2. Quotient Law of Logarithms
loga(m/n) = logam - logan
a>0, m>0, n>0, a≠1

Proof:
let m = ax
let n = a
y

Now...turn these into logarithms:
logam = x
log
an = y

So...
m/n = ax / ay
m/n = a
x-y
loga(m/n) = x-y
loga(m/n) = logam - logan

Example:
Evaluate... log3405-log35
log3(405/5)
log381
3x=81
3x=34
x=4



3. Power Law of Logarithms
loga(nm) = mlogan

Proof:
let n=ax

therefore:
logan=x

nm = (ax)m
nm=ax*m
loga(nm)=x*m
loga(nm)=mlogan

Example:
log394
=4log39
=4log332
=4(2)log33
=4(2)(1)
=8

Today...we also learned about the Change of Base Formula:
  • it helps us solve problems like log210
log210
2x=10
?!?!??!?!?!?!??!!
WAIT A SECOND...There's still a way to solve it...
We shall use the Change of Base Formula!

logbX=logaX/logab

log210=log10/log2 (assume common logarithms...a=10)
now use your calculator:
log210=3.32 (approximate)
23.32=10

If you still don't understand...watch this video =)


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