Properties of Logarithms
1
Learning Objectives
1. Work with the properties of logarithms
2. Write a logarithmic expression as a sum or
difference of logarithms
3. Write a logarithmic expression as a single
logarithm
4. Evaluate logarithms whose base is neither 10
nor e
2
Properties
Exponential
Logarithmic
ar as = ar+s
logr ∙ s = logr + logs
ar/as
=
ar-s
logr/s = logr - logs
=
ars
logrs = s∙logr
(ar)s
ar br = (ab)r
log ar br = r∙loga + r∙logb
3
1
Example
Expand
log  x  x  1 
logr ∙ s = logr + logs
log  x  x  1   log x  log  x  1
Log of product  sum of logs
4
Example
 x  1
log 
 x 
Expand
logr/s = logr - logs
 x  1
log 
 log  x  1  log x
 x 
Log of quotient  difference of logs
5
Example
logrs = s∙logr
Expand
x
log  x  1 


x
log  x  1   x  log  x  1


Log of power  power times log
6
2
Properties
Exponential
ay
Logarithmic
ax
=
a1
=a
logbb = 1
a0
=1
logb1 = 0
a-r = 1/ar
blogb x  x
x=y
logx = logy
x=y
log a-r = -r ∙loga
logb b x  x
7
Example
Solve
log  x  3  log 5
We use logx = logy  x = y to solve
log  x  3  log 5
x3 5
x2
2
8
Example
Solve
log  x  3  0
We use logb1 = 0
log  x  3  0
log  x  3  log1
x 3 1
x  2
2
9
3
Example
log  x  3  1
Solve
We use logbb = 1 to solve
log  x  3  1  log10
log  x  3  log10
x  3  10
x7
7
10
Example
log  x  3  2
Solve
We use logbb = 1 to solve
log  x  3  2 2 log10  log100
log  x  3  log100
x  3  100
x  97
97
11
Properties
Base 10
Base e
log10  1
ln e  1
log1  0
ln1  0
log10 x  x
ln e x  x
10log x  x
eln x  x
12
4
Remarks
We must learn how to use these properties to
•
solve logarithmic and exponential equations
•
expand logarithmic expressions
•
collect logarithmic expressions
13
Example
3
Expand log
3x  1
5x4
Using logr/s = logr - logs
 log 3 3x  1  log 5x4
Using logr ∙ s = logr + logs
 log 3 3x  1  log 5  log x4
Using logrs = s∙logr
1
 log  3x  1  log 5  4log x
3
14
Example
Rewrite as a single logarithm
4log3 x  3log3  x  1
Using logrs = s∙logr
Using logr ∙ s = logr + logs
 log3 x 4  log3  x  1
3
 log3 x 4  x  1
3
15
5
Domain
The domain of logb(x) is all x>0
When asked to find the domain of a log
function, simply set the argument greater than
zero and solve
16
Example
Find domain
f  x   log3  x  2 
x2  0
x2
D  f    2,  
 x2
f  x   log 2 

 x 3 
x  2  0  x  2
Example
Find domain
x2
0
x 3
x 3  0  x  3
3

0
2
3  2 1

0
3  3 6

02 2

0
0  3 3
4
3

42 6
 0
43 1
(-∞ , 2) U (3, ∞)
6
Example
f  x   log3 x  2
Find domain
Clearly x  2  0
and x  2  0 when x  2
D  f    , 2 
Example
f  x   ln  x  2 
Find domain
 x  2
2
 2,  
2
0
Clearly (x - 2)2 = 0 when x = 2
and is positive for all other x
D  f    , 2 
2
 2,  
Change of Base Formula
If b  0, b  1, a  0, a  1, and x  0,
then log a x 
logb x
logb a
Conversion to base 10
Conversion to base e
log x
log a
ln x
log a x 
ln a
log a x 
21
7
Example
Evaluate log 3 8
to 3 decimal places
0.946
Example
22
Express as a single log:
1
3log5 3  2log5 2  log5 4
2
 log5 3  log5 2  log5 4
3
2
 log5  27  4  2 
Example
12
1


 log5  33  22  4 2 


= log5216
 2 x2 z 
Expand terms: ln 

3
 y 
 ln 2  ln x2  ln z  ln y3
1
 ln 2  2ln x  ln z  3ln y
2
8
2
Collect terms: 3ln x  ln y
3
Example
 x3
 x3 


ln
23 
 3 y2
y 





= ln x3  ln y 2 3  ln 
Example Condense
1
3log a x  log a  2 x  1  log a  x  1
4
 log a x3  log a  2 x  1 4  log a  x  1
1
 log a x3  log a 4 2 x 1  log a  x 1


 log a x3 4 2 x  1  log a  x 1
 log a
x3
2x 1
x 1
4
Write as the sum and/or
difference of logarithms.
Express all powers as factors
Example
log a
log a
x2 x  3
 x 1
2
x2
x3
 x  1
2
2
 log a  x 2 x  3   log a  x  1
 log a x2  log a x  3  log a  x  1
2
1
 2log a x  log a  x  3  2log a  x  1
2
9
Exponential Function Change of Base
y  a bx
Changing base to exponential
Take ln of each side
ln y  ln  a b x 
Use properties of logs
ln y  ln a  x ln b
Take exponential of each side
y  eln a  x ln b
Use properties of
exponentials
y  eln a e x ln b
28
Exponential Function Change of Base
y  eln a e x ln b
Simplify
y  ae
a b x  ae
ln b  x
ln b  x
29
Example
Change to base e
y  150 1.04 
Use
x
a b x  ae lnb x
y  150e
ln1.04  x
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