Transcript UNIT3 OUTCOME3 - Greenfaulds High School

Laws of Logarithms
Proofs
(I)
loga(xy) = logax + logay
(II)
loga(x/y) = logax - logay
(III)
logaxp = p logax
Let
then
m = logax
x = am
and
and
(I)
xy = am X an
so
xy = am+n
n = logay
y = an
so
loga(xy) = logaam+n
so
so
loga(xy) = m + n
loga(xy) = logax + logay
(II)
so
m  an
=
a
y
x/ = am-n
y
x/
so
loga(x/y) = logaam-n
so
loga(x/y) = m - n
so
loga(x/y) = logax - logay
(III)
xp =(am)p
so
xp = amp = apm
so
so
logaxp = logaapm
logaxp = pm
so
logaxp = plogax
Ex1
log315 - log35 = log3(15/5)
(by law 2)
= log331
= 1
Ex2
4log42 = log424
= log416
= log442
= 2
Alt
4log42 = 4 x log44
= 4 x log441/2
= 4 X 1/2
= 2
(by law 3)
Ex3
1/
2log616
+ 2log63
= log6161/2 + log632
(law3)
= log616 + log69
= log64 + log69
= log6(4 X 9)
= log636
(law1)
= log662
=
2
If logx = logy then x = y
If logx > logy then x > y
If logx < logy then x < y
Ex4
If logam = 7logat + loga5 then show that m = 5t7
**********
logam = 7logat + loga5
logam = logat7 + loga5
(law3)
logam = loga (t7 X 5)
(law1)
logam = loga5t7
m = 5t7
Which is bigger
500499 or
499500 ?
(NB: value of each is too large for calculator to show)
*******
Using law3
log10 500499 = 499 log10 500 = 1346.8
Ex5
log10 499500 = 500 log10 499 = 1349.1
log10500499 < log10499500
so 499500 > 500499