section 5.3 properties of logarithms advanced algebra
TRANSCRIPT
Section 5.3 Properties of Logarithms
Advanced Algebra
Properties of Logarithms
logb1 = 0 Anything raised to the 0 power is 1.
logbb = 1 Anything raised to the 1st power is that anything
logbbx = x Think about as exponent: bx = bx
Rewrite as a log: logbx = logbx
xb xb log
Common and Natural Log Properties
log 1 = 0
log 10 = 1
log 10x = x
10log x = x x > 0
ln 1 = 0
ln e = 1
ln ex = x
eln x = x if x > 0
Simplifying
4log4
1
310log
31010log
3
Simplify Using Properties
3 2ln e3
2
lne
32
4log3 22
64
4log3 223
2 4log234
)4log( 2
10 x
42 x
More Simplifying
x310log
x3
elnln
1ln
0
More Simplifying
01.log
210log
2
25log5
5log5
1
27log3
21
27log3
21
33 3log
23
3log3
2
3
More Simplifying
35 625log
31
625log5
31
45 5log
34
5log5
3
4
3 More Properties
Product Rule logbMN = logbM + logbN
Quotient Rule
Power Rule
NMN
Mbbb logloglog
MpM bp
b loglog
Write as a sum or difference of logs
4 32log yu
432log yu
43
loglog 2 yu
yu log43log2
5log
4
2
x
5loglog 24
2 x
5loglog4 22 x
283
12log
2
2
xx
xx
74
34log
xx
xx
74log34log xxxx
7log4log3log4log xxxx
7log4log3log4log xxxx
Expand as a sum and difference of logs
Expand the log into a sum or difference of logs
441
log22
2
xxx
xx
22 21
1log
xx
xx
22 2log)1log()1log(log xxxx
2log2)1log()1log(log 2 xxxx
Write the difference as a single log
ba ln3ln2
1
3lnln 21
ba
3
21
lnb
a
3lnb
a
Write the sum as a single log
zyx 333 log8log2log
21
32
33 log8loglog zyx 4
32
33 logloglog zyx 42
3log zxy
To use the product or quotient rules of logs, remember, the bases must be the same.
Combine into a single log
zyx ln33lnln2
32 ln3lnln zyx 3
2
ln3
ln zy
x
y
zx
3ln
32
Condense to a single logarithm
4ln6ln3
1)9ln(
4
1 22 xxx
4ln6ln)9ln( 31
41 22 xxx
4ln6ln9ln 3 24 2 xxx
46
9ln
3 2
4 2
xx
x
Evaluate log45
b
MMb log
loglog
y5log4
54 y
5log4log y
5log4log y
4log
5logy
16.1y
b
MMb ln
lnlog
Change of Base Formula
Evaluate
5log3
1
31log
5log
46.1