log properties ­ expand and condense.notebook
November 29, 2010
Three properties:
Let a ≠1, and let n be a real number. If u and v are positive
real numbers, the following properties are true:
1. log
a
(uv) = log
2. log
a
(u/v) = log
3. log
a
un = n log
u + log
a
a
a
u - log
a
a
v
v
u
Nov 24­7:33 AM
Some examples: Nov 24­7:36 AM
1
log properties ­ expand and condense.notebook
November 29, 2010
Condense:
1) log3x + log3(4x)
2) log10100x ­ log10xy
3) log28x3 ­ log2x2 + log2(y/x)
Nov 17­7:29 AM
Solve: log10(x + 2) ­ log10(x ­ 1) = 1
Nov 17­7:33 AM
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log properties ­ expand and condense.notebook
November 29, 2010
PROOFS!
3) loga(u)n
logbase(answer) = exponent
let u = ax
let M = logau
aM = u
(aM)n = un
anM = un
loga(u)n = nM
loga(u)n = n(logau)
.
Nov 30­9:03 AM
1) loga(uv)
let u = ax
let v = ay
⇒ logau = x
⇒ logav = y
uv = axay
= ax+y
loga(uv) = x + y
loga(uv) = logau + logav
Nov 30­9:10 AM
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log properties ­ expand and condense.notebook
November 29, 2010
2) loga(u/v)
let u = ax
let v = ay
⇒
⇒
logau = x
logav = y
Nov 30­9:18 AM
Using those same three properties we can also condense logarithmic expressions: (this will be particularly helpful when we get into solving equations by the end of the week)
Condense: log3x + log3y
Condense: log5m ­ log5n
Condense: 3log10p + (1/2)log10q
.
Nov 30­2:49 PM
4
log properties ­ expand and condense.notebook
November 29, 2010
On white boards:
Condense: (1/3)log 7 x + log 7 (x + 2)
Condense: 4 lnx ­ 3 lny Condense: 2 ln8 ­ 3 ln4
.
Nov 30­2:52 PM
Proof of the change of base formula!!
Who remembers the change of base formula?
logba =
log10a
log10b
or lna
lnb
To help this proof make sense, we first look at a couple examples:
3log39 = ?
4log416 = ?
So, blogba = ?
take the log10 of both sides:
log10(blogba) = log10a
bring down the exponent using the third property:
logba(log10b) = log10a
divide both sides by log10b:
logba = log10a
log10b
.
Nov 30­2:56 PM
5
log properties ­ expand and condense.notebook
November 29, 2010
Nov 16­4:05 PM
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